What is the qualitative description of a covariant derivative and the Christoffel symbol?
Both the covariant derivative and the Christoffel symbols are formalizations of the concept of a linear connection on a manifold, which I'll just refer to as "connection" from now on since it won't be ambiguous. Once you understand why a connection is needed, it becomes easy to understand both the covariant derivative and the Christoffel symbols.
At every point on a manifold, there is a set of possible directions you can travel in. Combining such a direction with a magnitude gives you something called a tangent vector. They're so named because when the manifold is embedded in Euclidean space, the tangent vectors can be represented as vectors that are tangent to the manifold at a given point when one point along the vector is placed at the given point. [Example: On the surface of the Earth, "north 5 km" and "east 2 km" are tangent vectors, as well as every other combination of a compass direction and a distance. But "up 5 km" would not be a tangent vector.]
Tangent vectors at two different points are unrelated to each other. To convince yourself of this, grab a sheet of paper and crumple it up. Then choose two points, A and B, on the crumpled up piece of paper, not right next to each other. Then imagine yourself as an ant standing at point A, facing some direction. Which direction tangent to point B is the "same" direction as your current direction? Keep in mind that the terrain at point B may be slanted differently from the terrain at point A, and there is no such thing as "north" here. Of course, you could try walking over to point B, and seeing which direction you then face---but it still depends on which path you take.
In order to identify two tangent vectors at different points on the manifold, we need to connect the two tangent spaces. This is what a connection does, as I'm sure you've already guessed. Now we're ready:
Qualitative description of a connection.
Suppose a smooth curve is given between two points on a manifold, A and B. Pick a tangent vector at A. We can ask what happens to this tangent vector as it moves along the curve from A to B. It should have some direction at each point along the curve. If you know how to assign directions at each point along an arbitrary curve, between arbitrary endpoints A and B, for an arbitrary tangent vector initially located at point A, the collection of information you know about the manifold that lets you do this is a connection. We say that a connection lets us execute parallel transport.
A connection must satisfy certain "common sense" rules. If you double the length of the initial vector, or generally multiply by k, the transported vector should be doubled at each point too, or multiplied by k. Likewise, if you add together two vectors and then transport them, you should get the same transported vectors along the way as if you had transported the two vectors separately and added them along the way. These two properties are called linearity. Also, suppose your curve from A to B passes through intermediate points P and Q. You have a rule for transporting vectors from A to B along a given curve, and you also have a rule for transporting vectors from P to Q along a given curve. If a curve from A to B passes through P and Q, then the results of transporting the vector along the part of the path between P and Q should be consistent with what you would get if you just started at P and ended at Q. In other words, how the transport works shouldn't depend on what's a distance behind you or a distance ahead of you. You can say that the connection only gives you local information.
As a consequence of this locality property, what the connection really tells you is how to perform infinitesimal transport of a vector, that is, from a given point to another point that is infinitesimally close to it. If you can do that, you can reconstruct the result of transporting the vector along some curve by doing an infinite number of infinitesimal steps. You will notice that this is the same principle that integration is based on, and integration can be made fully formal, so while I have given a very, ahem, qualitative description of a connection, it is possible to formalize it completely with calculus, like we can do with integrals.
The Christoffel symbols and the covariant derivative are both formal incarnations of the concept of a connection. It shouldn't be surprising that they're closely related to each other.
The covariant derivative answers the following question: suppose we have a vector field on a manifold, and you choose some path between two points A and B. What is the derivative of the vector field along this path? To see why this captures the essence of a connection, notice that in order to calculate the derivative of some quantity along a given path, we have to take two points on that path, and consider the value of the quantity at each of the two points, subtract the values, and then divide by the distance between the two points. In doing so, we are comparing or identifying two vectors at different points. Indeed, if the value we get for the derivative is zero, it means that we consider the vector field to be "unchanging" along this curve, or that the image of the vector field along the given curve is consistent with parallel transport. A derivative operator is also required to be linear and local. So if you have a connection, you can define a covariant derivative, and vice versa.
The Christoffel symbols are the coefficients you would use in order to do parallel transport directly in a given coordinate frame, that is, they tell you exactly how much each coordinate of a given tangent vector at a given point has to change by, as you infinitesimally transport that given vector along a given direction to another (infinitesimally close) point. To do parallel transport along a curve, you would take the limit of compounding such infinitesimal transportations as the step size goes to zero. The Christoffel symbols are a linear operator on the initial tangent vector and the infinitesimal tangent vector between the initial point and the final point, so both linearity and locality are satisfied.
The covariant derivative can be defined in a given coordinate frame using the Christoffel symbols. Take the ordinary directional derivative and then subtract a correction term that results from the infinitesimal parallel transport you need to perform in order to connect two different (but infinitesimally close) tangent spaces. This correction term will of course be proportional to the Christoffel symbols. The Christoffel symbols can likewise be defined in a given coordinate frame using the covariant derivative: just write down the formula for the covariant derivative and then read off the coefficients of the correction term. So the two are really equivalent, and I realize that makes it confusing sometimes when people talk about connections. Some people will talk as though the Christoffel symbols are the connection, but then shift gears when you ask them to define the connection formally, and talk about the covariant derivative. Based on this usage, it seems to make more sense to say that the connection is an abstract concept, and that the Christoffel symbols and the covariant derivative are both "incarnations" of this concept, and therefore essentially the same thing.
Finally, note that there are an infinite number of possible connections on every given manifold (and therefore an infinite number of different covariant derivatives, each with its own distinct set of Christoffel symbols in each coordinate frame). But if each tangent space is equipped with a symmetric bilinear form, called the metric, then there is one unique connection, called the Levi--Civita connection, that is singled out by the metric. We can then speak of the covariant derivative for that manifold, or the Christoffel symbols. I won't discuss this further here. Read here: What is an intuitive explanation of the Levi-Civita connection?
At every point on a manifold, there is a set of possible directions you can travel in. Combining such a direction with a magnitude gives you something called a tangent vector. They're so named because when the manifold is embedded in Euclidean space, the tangent vectors can be represented as vectors that are tangent to the manifold at a given point when one point along the vector is placed at the given point. [Example: On the surface of the Earth, "north 5 km" and "east 2 km" are tangent vectors, as well as every other combination of a compass direction and a distance. But "up 5 km" would not be a tangent vector.]
Tangent vectors at two different points are unrelated to each other. To convince yourself of this, grab a sheet of paper and crumple it up. Then choose two points, A and B, on the crumpled up piece of paper, not right next to each other. Then imagine yourself as an ant standing at point A, facing some direction. Which direction tangent to point B is the "same" direction as your current direction? Keep in mind that the terrain at point B may be slanted differently from the terrain at point A, and there is no such thing as "north" here. Of course, you could try walking over to point B, and seeing which direction you then face---but it still depends on which path you take.
In order to identify two tangent vectors at different points on the manifold, we need to connect the two tangent spaces. This is what a connection does, as I'm sure you've already guessed. Now we're ready:
Qualitative description of a connection.
Suppose a smooth curve is given between two points on a manifold, A and B. Pick a tangent vector at A. We can ask what happens to this tangent vector as it moves along the curve from A to B. It should have some direction at each point along the curve. If you know how to assign directions at each point along an arbitrary curve, between arbitrary endpoints A and B, for an arbitrary tangent vector initially located at point A, the collection of information you know about the manifold that lets you do this is a connection. We say that a connection lets us execute parallel transport.
A connection must satisfy certain "common sense" rules. If you double the length of the initial vector, or generally multiply by k, the transported vector should be doubled at each point too, or multiplied by k. Likewise, if you add together two vectors and then transport them, you should get the same transported vectors along the way as if you had transported the two vectors separately and added them along the way. These two properties are called linearity. Also, suppose your curve from A to B passes through intermediate points P and Q. You have a rule for transporting vectors from A to B along a given curve, and you also have a rule for transporting vectors from P to Q along a given curve. If a curve from A to B passes through P and Q, then the results of transporting the vector along the part of the path between P and Q should be consistent with what you would get if you just started at P and ended at Q. In other words, how the transport works shouldn't depend on what's a distance behind you or a distance ahead of you. You can say that the connection only gives you local information.
As a consequence of this locality property, what the connection really tells you is how to perform infinitesimal transport of a vector, that is, from a given point to another point that is infinitesimally close to it. If you can do that, you can reconstruct the result of transporting the vector along some curve by doing an infinite number of infinitesimal steps. You will notice that this is the same principle that integration is based on, and integration can be made fully formal, so while I have given a very, ahem, qualitative description of a connection, it is possible to formalize it completely with calculus, like we can do with integrals.
The Christoffel symbols and the covariant derivative are both formal incarnations of the concept of a connection. It shouldn't be surprising that they're closely related to each other.
The covariant derivative answers the following question: suppose we have a vector field on a manifold, and you choose some path between two points A and B. What is the derivative of the vector field along this path? To see why this captures the essence of a connection, notice that in order to calculate the derivative of some quantity along a given path, we have to take two points on that path, and consider the value of the quantity at each of the two points, subtract the values, and then divide by the distance between the two points. In doing so, we are comparing or identifying two vectors at different points. Indeed, if the value we get for the derivative is zero, it means that we consider the vector field to be "unchanging" along this curve, or that the image of the vector field along the given curve is consistent with parallel transport. A derivative operator is also required to be linear and local. So if you have a connection, you can define a covariant derivative, and vice versa.
The Christoffel symbols are the coefficients you would use in order to do parallel transport directly in a given coordinate frame, that is, they tell you exactly how much each coordinate of a given tangent vector at a given point has to change by, as you infinitesimally transport that given vector along a given direction to another (infinitesimally close) point. To do parallel transport along a curve, you would take the limit of compounding such infinitesimal transportations as the step size goes to zero. The Christoffel symbols are a linear operator on the initial tangent vector and the infinitesimal tangent vector between the initial point and the final point, so both linearity and locality are satisfied.
The covariant derivative can be defined in a given coordinate frame using the Christoffel symbols. Take the ordinary directional derivative and then subtract a correction term that results from the infinitesimal parallel transport you need to perform in order to connect two different (but infinitesimally close) tangent spaces. This correction term will of course be proportional to the Christoffel symbols. The Christoffel symbols can likewise be defined in a given coordinate frame using the covariant derivative: just write down the formula for the covariant derivative and then read off the coefficients of the correction term. So the two are really equivalent, and I realize that makes it confusing sometimes when people talk about connections. Some people will talk as though the Christoffel symbols are the connection, but then shift gears when you ask them to define the connection formally, and talk about the covariant derivative. Based on this usage, it seems to make more sense to say that the connection is an abstract concept, and that the Christoffel symbols and the covariant derivative are both "incarnations" of this concept, and therefore essentially the same thing.
Finally, note that there are an infinite number of possible connections on every given manifold (and therefore an infinite number of different covariant derivatives, each with its own distinct set of Christoffel symbols in each coordinate frame). But if each tangent space is equipped with a symmetric bilinear form, called the metric, then there is one unique connection, called the Levi--Civita connection, that is singled out by the metric. We can then speak of the covariant derivative for that manifold, or the Christoffel symbols. I won't discuss this further here. Read here: What is an intuitive explanation of the Levi-Civita connection?