If you just want a classical picture over the complex numbers, the objects lying over the cusp points are Néron polygons equipped with some extra structure.  To make a Néron $n$-gon, you take $\mathbb{Z}/n\mathbb{Z} \times \mathbb{P}^1$ ($n$-copies of the 2-sphere), and glue the components into a circular chain, with 0 in one sphere glued transversely to $\infty$ in the next.  The free action of $\mathbb{Z}/n\mathbb{Z}$ is part of the structure of the $n$-gon.

The nature of the extra structure depends on the moduli problem describing the modular curve.  Here are some examples:

 - A Néron 1-gon is a nodal cubic curve, so the object lying over the cusp of $X(1)$ is a nodal projective line equipped with a point in the smooth locus.  Silverman gives a description of an analytic neighborhood of the cusp in his Advanced Topics book, in the section on the Tate curve.  There is a more formal description in chapter 8 of Katz-Mazur.

 - For $X_0(N)$, the smooth locus parametrizes elliptic curves equipped with a cyclic subgroup of order $N$, and for each $m|N$ one has cusps that parametrize $m$-gons equipped with a distinguished point (the identity element) and a cyclic subgroup of order $N$ in the smooth locus.

 - For $X_1(N)$, the smooth locus parametrizes elliptic curves equipped with a point of order $N$, and for each $m|N$ one has cusps that parametrize $m$-gons equipped with a distinguished point (the identity element) and point of order $N$ in the smooth locus.

 - For $X(N)$, the smooth locus parametrizes elliptic curves equipped with an isomorphism from $(\mathbb{Z}/N\mathbb{Z})^2$ to the $N$-torsion, and for each $m|N$ one has cusps that parametrize $mN$-gons equipped with an isomorphism from $(\mathbb{Z}/N\mathbb{Z})^2$ to the $N$-torsion.

For each of these moduli problems, there are explicit formulas for the precise number of cusps for each $m$.  For $X_0(N)$, you can find it in Shimura, section 1.6.  For the others I think it isn't too hard to derive.

As others mentioned, Deligne-Rapoport's compactified moduli problems are described using the notion of generalized elliptic curve, which is a flat proper family of connected curves of arithmetic genus 1 and at most nodal singularities, equipped with a distinguished section and a "group law".  Unlike the smooth case, the group law is a necessary extra datum here because we wouldn't have a uniquely defined $\mathbb{Z}/n\mathbb{Z}$-action on the $n$-gon over a cusp without it.