A Fibonacci group is defined as the group with the presentation $$ F(2,m) = \langle a_1, a_2,\ldots, a_m \mid a_i a_{i+1} = a_{i+2} \rangle $$ where the indexes are reduced modulo $m$.
Can someone describe how this a group ? Please post some other references.
Let $F$ be the free group on $\{a_i\}_{i=1}^m$. Quotient out by the normal subgroup $N$ generated by the relations $a_ia_{i+1}=a_{i+2}$ (i.e., the elements (or "relators") $a_ia_{i+1}a_{i+2}^{-1}$), taking the subscripts modulo $m$. The resulting group $G\cong F/N$ is the group defined by your group presentation.
Such a description is typical (and in fact a definition of) a group presentation.
For a reference, you can't get more comprehensive than this survey of Fibonacci groups.