To add to what others say, the equation of a line in two dimensions is $y=mx+c$ - as I was taught it. As an operator this is a stretch (by $m$) and a translation (by $c$). Although translations are important (e.g. in dynamics - the equations of motion), and seem obviously to be linear in character, it turns out that it is very useful to have a concept of linearity which applies when the origin is fixed - i.e. we ignore translations, or they are unimportant in the context we are thinking about.
This idea turns out to be useful beyond the immediate world of vector spaces (tensor products, bilinear and quadratic forms, modules over a ring, linear differential equations, representation theory etc).
Basically a function $f$ is said to be linear if there is a system of "scalars" around (like a field or a ring), and if $\lambda$ and $\mu$ are arbitrary scalar factors then $f(\lambda a + \mu b) = \lambda f(a) + \mu f(b)$. That would be a basic definition to go with your "linear algebra" tag.
As other answers have suggested, there are other uses of the word "linear" in mathematics - there have been several questions on this site asking for clarification - it isn't always clear, and you just have to get used to it. The word affine is sometimes used to make it clear that translations are included.