I understand that the modular group is the group of linear fractional transformations generated by $z \mapsto z+1$ and $z \mapsto \frac{-1}{z}$. I am studying the book Sphere packings, lattices and groups by Conway and Sloane and have come across the subgroup of this group which is generated by $z \mapsto z+2$ and $z \mapsto \frac{-1}{z}$. I have looked at many books on modular forms and can't seem to find any information about this group unless it has just been presented in a different way and I've missed it! What is the name of this group?
How would one define modular forms in this subgroup?
(EDIT) How would one compute the cusps of this subgroup, identified below?
This is the Hecke triangle group with angles $(\pi/2, 0, 0)$. It's a congruence subgroup of $PSL_2(\mathbf{Z})$ of index 3 and level 2.
PS: As for the last sentence of your question (edited in later): there is a standard definition of modular forms of level $\Gamma$ and weight $k$, for any finite-index subgroup $\Gamma$ of $PSL_2(\mathbf{Z})$ and any even integer $k$. See e.g. Diamond + Shurman's book. This applies perfectly well to your group.