What is a tangent space, anyway?

My site is called the Tangent Space. This is something of a pun, on the fact I am prone to tangents and parentheticals. But actually, the tangent space is a concept in mathematics. Tangent spaces come up in both differential and algebraic geometry. They’re easier to understand in differential geometry though, so that will be the focus of this post. [1] There are actually several ways to define…