Q:

Problem with the concept of connection

I've been told that there is only a canonical way for doing the vertical subspace of the tangent bundle of a manifold and in order to do the horizontal subspace you need a connection. These are very elementary question, but I've just passed over them during my studies so I guess it is a good time to ask:
1 - I have no intuition whatsoever about a connection. What intuition is there to have?
2 - Why can't I simply take the orthogonal complement in each fiber to produce the "horizontal subspace"? (Answered).

A:

I'll try to give an intuitive answer, and I'm aware that it might be imprecise in some points. For a more detailed answer, try and have a look at Choquet-Bruhat Y., DeWitt-Morette C., Analysis, manifolds and physics.
When you consider a tangent bundle, you are attaching "continuously" a vector space to every point in the manifold. This tells us that, at least locally, we can imagine the bundle as a copy of $\mathbb{R}^{2n}$ where the first coordinates (horizontal) are coordinates of the manifold, and the others are coordinates on the fibres ("vertical"). Hence, it is natural to think that a vector which is tangent $\textit{to the tangent bundle}$(!) can be decomposed in two different components, one along the fibre and one along the manifold. Locally everything is fine. 
Now, the issue is the following: pick a point $x\in M$ and a vector $v$ in the fibre over $x$ (i.e. tangent to the manifold at $x$). Move some steps away from $x$, to a point $y$ on the manifold. There is NO canonical way (in general) to say what happened to your vector $v$ when you changed your point. In other words, there is no natural "parallel transportation" of your vector. The problem arises because moving from $x$ to $y$ you might change the basis of the fibre, and directional derivatives behave pretty badly with coordinate changes.
A connection is a tool designed to solve this problem: it "connects" the fibres to one another, and tells you how tangent vectors are transformed if you move along a path on the manifold. This might take some time to sink in, because it's pretty far away from the formal definition of a connection. However, it is exactly what you need to tell apart the two different directions of a tangent vector of the tangent bundle: vectors which are "horizontal" are sent to zero by the connection, which instead transforms vectors in the fibre, which are "vertical".

