TITLE: Purpose of cusps
QUESTION [10 upvotes]: In the theory of of modular forms, there is the set of of cusps defined by $\mathbb{P}^1 (\mathbb{Q})= \mathbb{Q} \cup \{\infty\}$. For an subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$ of finite index, we call $\Gamma\backslash \mathbb{P}^1 (\mathbb{Q})$ the set of cusp classes.
For example, I know that $\text{SL}_2(\mathbb{Z})/\mathbb{P}^1 (\mathbb{Q})$ consists of one element, i.e. represented by $\infty$; Modular forms which are written as Taylor series and vanish at any belonging cusp are named cusp forms.
However, what is so important of these two definitions? Do they have 'special' properties, i.e. allow a geometric interpretation of modular forms? Which are the features they provide and in which contexts of modular forms I maybe re-encounter them?

REPLY [2 votes]: As Paul Garrett intimates in his answer, the notion of cuspform is fundamental, but can be subtle in its meaning and implications.  Here are a few different points of view that might help:

From the point of view of someone interested in the relationship between modular forms and algebraic number theory, Hecke eigenforms correspond to (certain) two-dimensional representations of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$.  Under this correspondence, the Eisenstein series correspond to reducible two-dimensional representations (i.e. those which are the sum of two characters), while the cuspidal eigenforms correspond to irreducible representations.

A different answer, related to quadratic forms: 
The $\theta$ series of a positive definite quadratic form will is a modular form.
 If we expand it in terms of a basis of Hecke eigenforms, it will have some Eisenstein contributions, which can be more-or-less computed explicitly, and then some cuspform contributions, which in general aren't describable by an explicit formula. 
However, we know that the Hecke eigenvalues, and hence the $q$-expansion coefficients, of a cuspform grow at a much slower rate than those of an Eisenstein series.  (There are elementary bounds, due to Hecke, tighter bounds,
due to Rankin, and then the ultimate Ramanujan--Petersson bound, proved by Deligne.)  Just from this, we can derive asymptotics as $n \to \infty$ for the number of representations of $n$ by a given quadratic forms.  (See e.g. Serre's Course in arithemtic for concrete examples.)  

In weight two, as Agol notes in his answer, cuspforms correspond to holomorphic differential forms on the modular curve, while all modular forms correspond to differential forms that are holomorphic except for possible simple poles at the cusps.

Just to tie in a little with Paul Garrett's answer, in the spectral theoretic point of view on automorphic forms (not something you see in the beginning literature on holomorphic modular forms that number theory students tend to read, but something that is usually introduced in books that take a more representation-theoretic veiw-point), cuspforms (on a given group, and on its Levi subgroups) are the basic building blocks in terms of which the rest of the spectral theory is developed.