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 $\newcommand{\F}{\mathcal F}H^i(X,\F)$ ) and other cohomologies.
1) 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]
2) Cartan's theorem: Given a topological space $X$ and a sheaf $\F$, assume there exists a basis of open sets $\mathcal{U}$, 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: $H^i(U,\F)=0$
Then the Cech cohomology of $\F$ on $X$ coincides with (Grothendieck) cohomology
3) 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.
4) If $X$ is paracompact and Hausdorff, 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]
5) 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.
6) 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.)
c) Torsten Wedhorn's quite detailed on-line notes, which prove 1) above (Theorem 9.16, p.92) and much, much more.
By the way, @Wedhorn is one of the two authors of a great book on algebraic geometry.
d) Ciboratu, Proposition 2.1 and Voisin's Hodge Theory and Complex Algebraic Geometry I, Theorem 4.47, page 109 , which both also prove 1) above.
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories with enough injective objects. Let me use the notation $D^+(\mathcal{A})$ and $D^+(\mathcal{B})$ to denote the stable $\infty$-categories whose homotopy categories are the (cohomologically bounded below) derived categories of $\mathcal{A}$ and $\mathcal{B}$, respectively (you can also consider unbounded derived categories, but the situation is a bit more subtle).
Let $\mathcal{C} \subseteq \mathrm{Fun}( D^{+}( \mathcal{A} ), D^{+}( \mathcal{B}) )$ be the full subcategory spanned by those functors which are exact, left t-exact, and carry injective objects of $\mathcal{A}$ into the heart of $D^{+}( \mathcal{B} )$. Then the construction
$$F \in \mathcal{C} \mapsto h^0 F|_{ \mathcal{A} }$$ determines an
equivalence from $\mathcal{C}$ to the category of left exact functors from $\mathcal{A}$ to $\mathcal{B}$. The inverse of this equivalence is "taking the right derived functor".
Consequently, one can answer your question as follows: given a functor of triangulated categories $G: hD^{+}(\mathcal{A}) \rightarrow hD^{+}(\mathcal{B})$, it arises as a right derived functor (of a left
exact functor of abelian categories) if and only if
a) The functor $G$ lifts to an exact functor of stable $\infty$-categories $D^{+}(\mathcal{A}) \rightarrow D^{+}(\mathcal{B})$ (anything that you build by composing derived functors will have this property).
b) The functor $G$ is left t-exact and carries injective objects of
$\mathcal{A}$ into the heart of $hD^{+}(\mathcal{B})$.
Best Answer
There's a long story that can be told here but I will try to be brief. In one sense, the answer is yes – you can certainly define cohomology and homotopy groups and so on for pretoposes and have them coincide with the classical definitions for Grothendieck toposes – but in another sense the answer is no – because you are essentially just embedding the pretopos into a suitable Grothendieck topos and reducing to that case.
Let $\mathcal{E}$ be a pretopos. That means $\mathcal{E}$ is a category with a terminal object, pullbacks, finitary coproducts, and coequalisers of internal equivalence relations, such that finitary coproducts are disjoint and preserved by pullback, and coequalisers of internal equivalence relations are effective and preserved by pullback. In short, $\mathcal{E}$ satisfies the exactness part of the Giraud axioms, with finitary coproducts eplacing infinitary coproducts. That in itself should be a powerful reason to believe that any finitary constructions that can be carried out in a Grothendieck topos can also be carried out in $\mathcal{E}$ with the same results. Indeed:
Proposition. Assuming $\mathcal{E}$ is small, there is a fully faithful embedding of $\mathcal{E}$ into a Grothendieck topos where the embedding preserves finite limits, finitary coproducts, and coequalisers of internal equivalence relations.
Proof. Regard $\mathcal{E}$ as a site where the covering sieves are those that contain a sieve generated by a finite family that is jointly strongly epimorphic, and take the topos of sheaves on this site. ◼
(If $\mathcal{E}$ is not small then go up to a universe where it is, or find a subpretopos that is small and contains the objects and morphisms you care about.)
Concretely, the category $\textbf{Ab} (\mathcal{E})$ of internal abelian groups in $\mathcal{E}$ is an abelian category (but not necessarily AB4 or AB4*, let alone AB5). So you can go on to define the category $\textbf{Ch} (\mathcal{E})$ of chain complexes in $\textbf{Ab} (\mathcal{E})$ and then the (unbounded) derived category $\mathbf{D} (\mathcal{E})$. What you do not get is the existence of enough injectives in $\textbf{Ab} (\mathcal{E})$ itself. Nonetheless, the definition of derived functors as (absolute, or at least pointwise) Kan extensions makes sense, and some derived functors can be constructed without injective resolutions. For example, although $\mathbf{R} \textrm{Hom}_{\textbf{Ch} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \mathbf{D} (\textbf{Ab})$ itself does not have an obvious construction, $H_0 \mathbf{R} \textrm{Hom}_{\textbf{Ch} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \textbf{Ab}$ always exists: you can directly check that $\textrm{Hom}_{\mathbf{D} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \textbf{Ab}$ works. Also, if $\mathcal{E}$ is small, then any functor $\textbf{Ch} (\mathcal{E}) \to \textbf{Ab}$ whatsoever admits a pointwise left Kan extension along $\textbf{Ch} (\mathcal{E}) \to \mathbf{D} (\mathcal{E})$... but it is unclear to me whether this is consistent with what $(\infty, 1)$-category theory would give.
Similarly (in some sense...), the category $\textbf{Kan} (\mathcal{E})$ of internal Kan complexes in $\mathcal{E}$ is a category of fibrant objects (in the sense of Brown) where the fibrations are the internal Kan fibrations and the weak equivalences are the internal weak homotopy equivalences. Thus the homotopy category $\mathbf{H} (\mathcal{E})$, obtained by localising $\textbf{Kan} (\mathcal{E})$ with respect to internal weak homotopy equivalences, is reasonable in the sense that there is a nice-ish formula for its hom-sets. The category $\textbf{Set}_\textrm{fin}$ of finite sets is the initial pretopos, so we get an induced functor $L : \mathbf{H} (\textbf{Set}_\textrm{fin}) \to \mathbf{H} (\mathcal{E})$. The homotopy type of $\mathcal{E}$ is "morally" a representing object for the functor $\textrm{Hom}_{\mathbf{H} (\mathcal{E})} (1, L {-}) : \mathbf{H} (\textbf{Set}_\textrm{fin}) \to \textbf{Set}$, where $1$ is the terminal object, but in practice this functor is rarely representable (even if $\mathcal{E}$ is a Grothendieck topos) so we are forced to make various tweaks like replacing $\textbf{Set}_\textrm{fin}$ with $\textbf{Set}_{< \kappa}$ (if $\mathcal{E}$ has coproducts of families of size $< \kappa$), or allowing more generalised notions of representability (e.g. pro-representability), or both.
(Actually, this phenomenon can already be seen at homotopy level 0 for Grothendieck toposes, so in some sense the difficulty does not (only) come from trying to work in higher homotopy levels or with general pretoposes instead of Grothendieck toposes.)