Let $G$ be a group. Consider an arbitrary equation given by $w(\vec{g})=e$, where $w: G^n \to G$ takes an $n$-tuple $(g_1,...,g_n)$ to some expression involving products of the $g_i$, their inverses and other elements of $G$.
Are there any good methods to determine whether such equation has a solution $\vec{g} \in G^n$ or not? For which classes of groups do they work? I'd appreciate references on related results.
If you are interested in finite simple groups, there is a whole raft of literature considering the related question of which words maps $w: G^n\to G$ are surjective. The answer is often yes, so this gives a strong affirmative answer to your question in this case.
In general you should search the literature for work by Shalev, Garion, Liebeck, O'Brien, Larson, Tiep (and many others; please forgive me as I know I've missed many people out!)
Specific references:
M. W. Liebeck and A. Shalev, Diameters of finite simple groups: sharp bounds and applications, Ann. of Math. 154 (2001) 383–406.
The main result here is that for a given non-trivial word $w$, every element of every sufficiently large finite simple group $G$ can be expressed as a product of $C(w)$ values of $w$ in $G$, where $C(w)$ depends only on $w$;
M. Larsen and A. Shalev, Word maps and Waring type problems, J. Amer. Math. Soc. 22 (2009) 437–466.
M. Larsen, A. Shalev and P. H. Tiep, The Waring problem for finite simple groups, Ann. of Math. 174 (2011) 1885–1950.
A. Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. 170 (2009) 1383–1416.
These three papers improve the $C(w)$ from the last bullet point to $2$. This is best possible - improving C(w) to 1 is not possible in general, as is shown by power words $x^n$ , which cannot be surjective on any finite group of order non-coprime to $n$.
There are a host of other results in the same direction, so I can't be exhaustive. I'll just mention one little paper I came across recently which is related and which I thought was pretty groovy.
Sebastian Jambor, Martin W. Liebeck and E. A. O’Brien, Some word maps that are non-surjective on infinitely many finite simple groups, Bull. London Math. Soc. 45 (2013) 907–910
This may be largely irrelevant to what you're interested in, but in case it helps:
Philip Hall, On a theorem of Frobenius, Proceedings of the London Mathematical Society, (2) 40 (1936), 468-501.
There are quite a lot of results in the literature about the number of commutators needed to express an element of the derived group, and the character table is useful here. These are generally rather more elementary than the results mentioned by Nick Gill, One which I like (which I think appears as an exercise in W.Burnside's book on Group Theory) is that an element $g \in G$ is expressible as a product of $k$ commutators if and only if $\sum_{\chi} \frac{\chi(g)}{\chi(1)^{2k-1}} \neq 0,$ where $\chi$ runs over the complex irreducible characters of $G.$ The special case $k =1$ was useful in the verification of Ore's conjecture (that every element of a finite non-Abelian simple group is a commutator) by Liebeck et al.
