A table algebra is an abstract setting for the study of products and decomposition of products of either irreducible ordinary characters or conjugacy classes of finite groups. Table algebras are introduced in this paper and their properties are studied. The results on table algebras are then translated into corollaries on products of irreducible characters and on products of conjugacy classes in finite groups. Thus, separate but analogous results on characters and on conjugacy classes are treated uniformly. Let \(A\) be an associative, commutative finite-dimensional algebra with 1 over the field of complex numbers together with a basis \(\mathbf B\) containing 1 and whose structure constants are nonnegative real numbers. \((A,\mathbf B)\) will be called a table algebra if it satisfies two conditions which we will not give here. The algebra of the complex class functions of a finite group together with the basis \(\text{Irr}(G)\) of the irreducible characters is a table algebra. A second example is the center of the group algebra of a finite group over the complex number field together with the basis consisting of the conjugacy class sums. Other examples are given in the paper. We give here one example of the many results obtained. Let \(G\) be a finite nonabelian simple group with \(k\) conjugacy classes, of which \(r\) are real classes. Let \(C\neq 1\) be a conjugacy class of \(G\) and \(\chi \neq 1\) an irreducible character of \(G\). Set \(m =(k^ 2 - (r - 1)^ 2)/2\). The known facts that \(C^ m = G\) and that \(\chi^ m\) contains all elements of \(\text{Irr}(G)\) as constituents are (both) proved here as a consequence of a more general result on so- called covering numbers of ``simple'' (a term that is defined in the article) table algebras.