What is the difference between a chart and a tangent space? To my lay-person mind, a chart is a one-to-one function that maps an area on a manifold to a euclidean space of equal dimension.
Then I understand a tangent space to be the space of vectors that are parallel to the manifold at a specific point. 
So does one use the appropriate charts to map points to the tangent space?
These seem like similar things to me (even though I am fairly certain they are not). Is the difference that any point on the manifold can be mapped to that specific tangent space whereas the chart only covers a subset of points in the manifold?
Or do I have two completely different ideas conflated?
Is the mapping of points from the manifold even related to the manifolds charts?
 A: Each point on a differentiable manifold has its own tangent space. Informally, you can think of the tangent space at a point as the space of tangent vectors to the surface at that point. But this informal picture only works if the manifold is embedded in some higher dimensional space.
A more abstract and more formal point of view is that tangent space at a point $P$ of a differentiable manifold $M$ consists of the derivatives of all curves on $M$ that pass through $P$. The derivatives exist because each map between $M$ and a chart is differentiable and invertible. So we can map a curve on $M$ to a chart, carry out calculus in the chart and "lift" the result back to $M$. The result is independent of the choice of chart, so it is a property of $M$ itself (at $P$).
It is not entirely obvious that we can create a vector space from the derivatives of the curves through $P$, but we can because the derivative is a linear operator. This vector space is then the tangent space at $P$, which we denote by $T_P$. The set of tangent spaces $T_P$ for all points $P$ on $M$ forms a vector bundle over $M$.
