"Linear" in "Linear Algebra" means "closed under addition". In physics a "linear system" is one that satisfies the superposition principle, which is just the physics way of saying closed under addition. This means that if S_1 and S_2 are two possible states of the system (i.e. two possible solutions to the equations of motion), then S_1 + S_2 is also a possible state of the system (i.e. it also solves the equations of motion). The same word, "linear", is also used to describe the equations of motion in this case; one says that the equations of motion are linear if their solutions satisfy this superposition principle. This use of the word simply generalizes the fact that if you add two points on a line, you end up with a point on the same line.
Linear techniques (e.g. Fourier transformation, perturbation theory, etc.) can be used to approximate the behavior of non-linear systems over sufficiently brief time periods, but most non-linear systems can only be "solved" numerically and display complicated chaotic behavior.
