Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written H^i(X,F) http://latex.mathoverflow.net/png?H%5Ei%28X%2CF%29 ) and other cohomologies.
If X is locally contractible then the cohomology of a constant sheaf coincides with singular cohomology. [This is Eric's answer, but there is no need for his hypothesis that open subsets be acyclic]
Cartan's theorem: Given a topological space X and a sheaf F, assume there exists a basis of open sets \mathcal{U} http://latex.mathoverflow.net/png?%5Cmathcal%7BU%7D, stable under finite intersections, such that the CECH cohomology groups for the sheaf F are trivial (in positive dimension) for every open U in the basis: H^i(U,\mathcal{F})=0 http://latex.mathoverflow.net/png?H%5Ei%28U%2C%5Cmathcal%7BF%7D%29%3D0 Then the Cech cohomology of F on X coincides with (Grothendieck) cohomology
Leray's Theorem: Given a topological space X and a sheaf F, assume that for some covering (U_i) of X we know that the (Grothendieck!) cohomology in positive dimensions of F vanishes on every finite intersection of the U_i's. Then the cohomology of F is already calculated by the Cech cohomology OF THE COVERING (U_i): no need to pass to the inductive limit on all covers. This contains Dinakar's favourite example of a quasi-coherent sheaf on a separated scheme covered by affines.
If X is paracompact, Cech cohomology coincides with Grothendieck cohomology for ALL SHEAVES
If you think this is too nice to be true, you can check Théorème 5.10.1 in Godement's book cited below [So Eric's remark that no matter how nice the space is, Cech cohomology would probably not coincide with derived functor cohomology for arbitrary sheaves turns out to be too pessimistic]Cohomology can be calculated by taking sections of any acyclic resolution of the studied sheaf: you don't need to take an injective resolution. This contains De Rham's theorem that singular cohomology can be calculated with differential forms on manifolds.
If you study sheaves of non-abelian groups, Cech cohomology is convenient: for example vector bundles on X ( a topological space or manifold or scheme or...) are parametrized by H^1(X, GL_r). I don't know if there is a description of sheaf cohomology for non-abelian sheaves in the derived functor style.
Good references are
a) A classic: Godement, Théorie des faisceaux (in French, alas)
b) S.Ramanan, Global Calculus,AMS graduate Studies in Mahematics, volume 65. (An amazingly lucid book, in the best Indian tradition.)