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 Zn 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, S1,R1 and S2 and R2. Is it the case that <S1|R1> is isomorphic to <S2|R2>?
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.