At each point in a manifold there is a vector space often called the tangent space at that point. When we think of a vector in a manifold we must generally be specific about where in the manifold and then say that the vector is in the tangent space there. There is generally no connection between vectors in tangent spaces at different points except a topological construct called a fiber-bundle that is about vectors at nearby points on the manifold being nearly equal. On a manifold an affine connection is a mathematical structure that allows parallel transport of vectors along a path thru the manifold.
If the path is a loop and we carry basis vectors around the loop then they may be changed. If so then the affine connection defines a curved space. This concept of curvature is more general and less symmetric than that computed from a metric.
A metric for a manifold determines an affine connection.
Relative to some coordinate system for the manifold, the Christoffel Symbol of the Second Kind describes an affine connection. A coordinate system already suggests an affine connection—don’t change the vector components as you move the vector. A different coordinate system, however, suggests a different connection. If a metric is clearly in sight then the Christoffel Symbol usually denotes the affine connection stemming from the metric. This may lead to some confusion.
A ‘small’ ball, by definition a sphere, in a connected manifold, can be carried about with a symmetric affine connection to define a metric. If the affine connection is symmetric then this sphere may rotate as it traverses a loop, but it will not expand or shrink.