How to define cusps of a subgroup of $SL_2(\mathbb{Z})$ and holomorphicity of a modular form at these cusps.

Question

I understand that when we define modular forms on a subgroup of the modular group we gain extra "boundary points" or cusps, on top of the point at $\infty$, where we require that said modular form is holomorphic. I am struggling to find a consistent definition for the cusps of such a subgroup, or for motivation for the extra holomorphicity condition at these cusps (what does it mean in practice). Anything that would help my understanding would be appreciated!