The Chow groups of a noetherian scheme $X$ are the analogs of the singular homology groups of a topological space.
Let $X$ be a noetherian scheme. One defines the $k$-th Chow group of $X$ as the quotient of the group $Z_k(X)$ of algebraic cycles of dimension $k$ by the subgroup of algebraic cycles rationally equivalent to zero:
The Chow ring is the graded ring which is the direct sum of the Chow groups, with multiplication being the intersection product.
More generally one can use any adequate equivalence relation $\sim$ (e.g. $\sim_{num}, \sim_{hom}, \sim_{alg}$) in place of rational equivalence, to get groups
Chow groups appear as the cohomology groups of motivic cohomology (see there for details) with coefficients in suitable Eilenberg-MacLane objects.
The canonical reference is
The original references are
Pierre Samuel?, Rational Equivalence of Arbitrary Cycles. American Journal of Mathematics, Vol. 78, No. 2 (Apr., 1956), pp. 383-400
Claude Chevalley, Les classes d’equivalence rationnelles I. Séminaire Claude Chevalley, 3 (1958), Exp. No. 2, 14 (on NUMDAM)
Claude Chevalley, Les classes d’équivalence rationnelle, II. Séminaire Claude Chevalley, 3 (1958), Exp. No. 3, 18 (on NUMDAM)
The most general treatment can be found in the The Stacks Project:
Informal lecture notes by Jacob Murre?:
A concise definition of the notion of Chow group and related concepts is in