What are vectors and how are they used in programming?

I'm familiar with the mathematical/physics concept of a vector as a magnitude and a direction, but I also keep coming across references to vectors in the context of programming (for example C++ seems to have a stl::vector library which comes up fairly frequently on SO).

My intuition from the context has been that they're a fairly primitive construct most often used to represent something along the lines of a variable length array (storing its size as the magnitude, I presume), but it would be really helpful if somebody could provide me with a more complete explanation, preferably including how and why they're used in practice.