A couple of further clarifications, to supplement the extensive answer by marguax and the many comments:
- Chevalley's influential 1955 paper was mainly concerned with finding a uniform approach to most of the known simple finite groups of Lie type (supplemented soon afterward by the introduction of twisted groups as well as the groups of Suszuki and Ree). The original "Chevalley group" was defined as a group of automorphisms of a certain Lie algebra, generated by unipotent elements; this group is simple over almost all fields (except a few very small ones). The Lie algebra here is obtained by first reducing mod $p$ a Chevalley $\mathbb{Z}$-basis of a simple Lie algebra over $\mathbb{C}$, then tensoring with a field of prime characteristic $p$.
A subtle point here is that the "Chevalley Lie algebra" obtained over an algebraically closed field is actually the Lie algebra of the corresponding simply connected algebraic group, whereas the Chevalley groups themselves are closer to the adjoint groups.
Anyway, Steinberg in his 1967-68 Yale lectures broadened the notion of "Chevalley group" by using other faithful Lie algebra representations, to include special linear groups and the like.
- In fact, there is a huge amount of literature about Chevalley groups over (commutative) rings. Here the ideal structure of the ring contributes to normal subgroup structure in the group, so the groups are typically non-simple. But they do come up naturally in algebraic K-theory, including the study of the congruence subgroup problem. N.A. Vavilov and many others have written extensively about such groups. There is less textbook literature along these lines, except for a few books on algebraic K-theory including one by Hahn and O'Meara. And the entire subject becomes quite technical.
ADDED: Concerning the more sophisticated viewpoint of Chevalley-Demazure group schemes, there is an important recent paper by Lusztig (not yet freely available online) here. The arXiv preprint is here.