TITLE: What is the difference between an automorphic form and a modular form?
QUESTION [49 upvotes]: This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it is. Many sources use the term in different ways.
Any classical holomorphic modular form for $\mathrm{SL}_2(\mathbb{Z})$ is called a modular form, and usually (but not always) so are modular forms for congruence subgroups. Often, "automorphic form" is used when one considers either other Fuchsian groups, forms on groups other than $\mathrm{GL}_2$, or non-holomorphic forms (such as Maass forms and real analytic Eisenstein series). Alternatively, Diamond and Shurman define an "automorphic form" in Section 3.2 as being like a modular form but possibly meromorphic instead of holomorphic.
As another example, Miyake's book Modular Forms writes on p.114, "Automorphic functions and automorphic forms for modular groups are called modular functions and modular forms, respectively," and his definition of "modular group" seems to coincide with that of congruence subgroup.
The Princeton Companion to Mathematics writes in section III.21, "automorphic forms, which are generalized versions of the classical analytic functions called modular forms [III.61]," but it does not specify what the generalization is. The book contains other, similar statements (in III.61, "And indeed, automorphic forms, which are generalizations of modular forms").
So what exactly is the* definition of modular form as opposed to automorphic form? Since there is likely no "right" answer, what I really want to know is what is the history and what are the different conventions and the relations between them.

REPLY [6 votes]: There are many aspects to the relation between modular forms and automorphic forms. First let me recall what a modular form of weight $2k$ for, say $SL(2,{\mathbb Z})$ is. This is a function on the upper half plane, which is $holomorphic$, has an equivariance property 
$$f(\frac{az+b}{cz+d})=(cz+d)^{2k}f(z)$$ for 
$\begin{pmatrix} a & b \cr c & d\end{pmatrix}\in SL(2,{\mathbb Z})$ 
 and at infinity (the "cusp" for $SL(2,{\mathbb Z})$ in the fundamental domain) has the Fourier expansion 
$$f(z)= \sum _{k=0}^{\infty} a(n)q^n,$$  where $q=e^{2\pi i z}$ for $z$ in the upper half plane. For example, the Ramanujan $\Delta$ is such a modular form. But, $1/\Delta (z)$, while it is still holomorphic on the upper halp plane, has its Fourier expansion at infinity starting from $q^{-1}$ and is hence not a modular form.  
We can convert a modular  form into a function on the group $SL(2,{\mathbb R})$ by taking the related function $F_f(g)= (ci+d)^{-2k}f(g(i))$. This is a function on $SL(2,{\mathbb R})$ on which the centre of the enveloping algebra acts by scalars (translating the holomorphy condition on $f$), is $SL(2,{\mathbb Z})$ invariant (translating the equivariance for the modular form $f$),  and has moderate growth on the fundamental domain for $SL(2,{\mathbb Z})$ in $SL(2,{\mathbb R})$. 
This can be generalised to any congruence subgroup of the modular group, and any weight etc, and you get $K$-finite , $Z$ finite functions on $G$, which are $\Gamma$ invariant. These are called automorphic forms. If you do this construction for the inverse of $\Delta$ you get most properties, but the moderate growth, and hence you do not get an automorphic form.  
On the other hand, any function on $G/\Gamma$ (say, $G$ semi-simple and $\Gamma$ an arithmetic group) which is $K$ finite, $z$ finite and has moderate growth (along with its derivatives ) on a fundamental domain, is defined to be an automorphic form. So even for $SL(2,{\mathbb R})$ there are many more automorphic forms than those arising from modular forms (those from modular forms generally have infinite component of the discrete series (this keeps track of the weight $k$ of the modular form)).