SHORT VAGUE DESCRIPTION OF MODULAR FORMS MOD P AND WHY WE CARE.
Modular forms (mod p) were initially defined by reduction modulo p of modular forms whose q-expansions are p-integral (i.e. rational numbers with the denominator not divisible by p). This point of view had considerable success (see, e.g., the theory of congruences developed by Serre and Swinnerton-Dyer), and continues to be widely used. The introduction of Grothendieck-style algebraic geometry in the theory of modular forms provided an intrinsic definition of modular forms (mod p), and more generally of modular forms with coefficients in an arbitrary abelian group. This definition is more general than the naive, reduction-based one, and there is considerable evidence that it is the appropriate one (a notable example is that of weight rules in Serre's conjecture).
Classical modular forms
Modular curves
Modular forms
Hecke correspondences and operators
Lifting to characteristic zero
Katz proved that all modular forms (mod p) of weight at least 2 can be lifted to characteristic zero (i.e. they are mod p reductions of modular forms over \mathbb{Z}). Mestre found many examples of weight 1 forms that cannot be lifted.
Computational aspects
Computing with mod p forms of weight at least 2 can be done by computing with characteristic zero forms, then reducing mod p.
Gabor Wiese has MAGMA packages and data for mod $p$ Hecke algebras and weight one modular forms over finite fields. The latter extends Mestre's computations.
Now that Serre's conjecture is a theorem of Khare, Wintenberger, Kisin, et al, the following question seems to be in line. For q a prime power and S a finite set of primes, there are finitely many odd continuous irreducible G_{\mathbb{Q}}\to GL_2(\mathbb{F}_q) which are unramified outside S; is it practical to list all of them?
Siegel modular forms
Siegel modular varieties
Siegel modular forms
Hecke correspondences and operators
It is not known for which weights all mod p Siegel modular forms can be lifted to characteristic zero; therefore, at the moment, computing with them needs to be done purely in positive characteristic. If one is only interested in computing the Hecke eigenvalues, it is possible to reduce this to a combinatorial problem involving isogeny graphs on the locus of superspecial abelian varieties.
Hilbert modular forms
Hilbert modular varieties
Hilbert modular forms
Hecke correspondences and operators
Dembélé and Diamond have data toward Serre's conjecture for Hilbert modular forms.