The smooth manifold is a place for vectors. At each point in the manifold there is a whole vector space whose dimension is the same as that of the manifold. This is called the tangent space at the point for one may imagine the manifold embedded in a larger flat space, where there would be a flat subspace tangent to each point of the manifold. Even when there is no known flat space in which to embed the manifold we speak of the tangent space, merely referring to those vectors found at the point of the manifold in question.

There is a tangent space for each point of the manifold and thus a nearby manifold point has its own tangent space. If it is very near we are tempted to identify vectors in the two tangent spaces just as we are tempted to compare wind velocity in two nearby cities, but when the cities are far apart it is not clear what it means to say the wind velocity is the same. Saying it is to the North at 15 km/hr in both cities may be meaningful, but only relative to the Earth’s conventional coordinate system. Manifolds generally lack such special coordinate systems. The mathematical concept of tangent bundles makes one topology of all of these tangent spaces.

A manifold may carry an affine connection by which vectors may be carried unaltered along some path thru the manifold. With an affine connection, a path between two points of the manifold establishes a linear transformation between the tangent spaces at those points.

A metric determines a symmetric affine connection, and a symmetric affine connection together with a sphere at some point of the manifold, determines a metric.

Here is a nice coordinate free presentation of the affine connection. Its terminology is at odds to what I use here. They start with the vector space of vector fields over the manifold. The connection between the two sets of ideas is not what first occurred to me. In the expression Γjki They reason about what they call a connection Ajki but in coordinate free notation. It is a field that satisfies the following equations.
Riemannian geometry texts note that the Christoffel symbols, sometimes written: Γijk, do not transform as tensors. On the other hand an affine connection on an affine manifold is. What gives?

The Christoffel symbol can be thought of as describing how a coordinate system twists and stretches as you move about on the manifold. It thus is about a particular coordinate system in two different ways:

The affine connection, on the other hand, speaks of no particular coordinate system but is, as any tensor representation, expressed in some coordinate system.

It is possible to express the twisting and stretching of one coordinate system in another coordinate system. It we keep the latter coordinate system unchanged then that is a tensor expression referring to some subject coordinate system.

A note on non-symmetric metric tensors.