\textit{D. M. Clark} and \textit{P. H. Krauss} [``Global subdirect products'', Mem. Am. Math. Soc. 210 (1979; Zbl 0421.08001)] define an algebra to be globally (Boolean) representable by a class of algebras \({\mathcal M}\) if it is isomorphic to the algebra of the global sections of a subdirect sheaf whose index space is compact and \(T_ 0\) and whose stalks belong to \({\mathcal M}\) or are trivial and at most one stalk is trivial. This paper presents general characterizations of global and Boolean representability of an algebra \(A\) by a class \({\mathcal M}\). Such characterizations are given in terms of properties of the congruence lattice of \(A\). A few simple new concepts are defined (i.e. compactness of a set of congruences, projective on \(A\), Chinese Remainder Theorems, etc.) which are sufficient to completely express the characterizations. One of the advantages of this approach is that since the characterizations are purely structural, one can see more clearly the sheaf-representational nature of a given algebra.