See Noether's excellent generators and relations for groups if you don't know about these things, or this will be gibberisher than usual.
A group G is *finitely presented* if it can be described by a finite set of generators S and a finite set of relations R: G=<S|R>.

If G is finite, then it is of course finitely presented: take e.g. S=G and G's multiplication table as R. But many infinite groups are also finitely presented. For instance **Z**^{n} is finitely presented. **Q** and **R**, on the other hand, are not finitely presented.

A finitely presented group can be generated by more than one set of generators and relations. For instance, **Z**=<a> (`a' is usually called "1" or "-1"). But we also have **Z**=<b,c | b+c=c+b, b+b+b=c+c> (`b',`c' are often called "2,3" or "-2,-3").

Generation of elements given S and R is (fairly) easily done by a computer (but doing it efficiently is *hard*, for much the same reasons as we proceed to discuss). Suppose we have 2 pairs of generators and relations, S_{1},R_{1} and S_{2} and R_{2}. Is it the case that <S_{1}|R_{1}> is isomorphic to <S_{2}|R_{2}>?

As an easy example ("easy" here is relatively speaking; it's actually somewhat tricky), can you prove the 2 representations of **Z** above indeed generate the same group?

The generators and relations are easily fed into a computer. But it turns out that this problem is *undecidable*: no computer program exists that can solve all instances of it.