A vector is a thing that can be added together and scaled in "the intuitive way." That is, for example, if a and b are numbers and v is a vector, then av + bv = (a+b)v (vector addition distributes over scalar multiplication). The prototypical example is the collection of arrows rooted at the origin on the 2D plane, where addition has a simple geometric interpretation (you put the tail of one vector at the tip of another, the resulting point is the new tip) and scaling is "stretching." But it really could be anything that adds and scales.

Not really.