What do mathematicians mean when they say "form"? As in differential form, modular form, quadratic form? 
I'm sorry if this is a really silly question.
 A: A main possibly non-intuitive usage of "form"  is as a somewhat particular type of map/function. 
Traditionally, the word function was used in a more restrained way and it was mainly used for real and complex functions, only. 
For example, classically in (real) functional analysis one would have: 


*

*A function would be a map from $\mathbb{R}$ to $\mathbb{R}$. 

*An operator would be a map from a space of functions to a space of functions. Example: the map "derivative" so $f \mapsto f'$. The term differential operator is still very common.  

*A form would be a map from a space of functions to $\mathbb{R}$, often a linear one. Example: definite integral, $f \mapsto \int_0^1 f(t)dt$.  
It is still common in this context and in linear algebra that forms (linear, bilinear, etc) map from a (vector) space to the reals (more generally the scalar field). 
Thus, a form is often a map from a  'complicated' domain (often some space) to a simpler co-domain (typically a field). 
The terminology differential form also goes under this umbrella.
And, while slightly less clear I'd argue quadratic form (and alike), too. 
That said, there are altogether different usages of the word "form" too though, as mentioned in the other answer, notably in the compound "normal form." But there the usage seems more in line with a common sense dictionary meaning of the word.  When there are several different yet equivalent ways to express something, then you can write it in one way  or in another way, in one form or in another form.  
But if the "form" is naturally seen as a map, frequently the co-domain will be 'simpler' than the domain.  
A: I'm not a Mathistorian, but... Likely it originally meant its English meaning of "appearance", and it still does in most usages. Quadratic forms have a very specific appearance, namely a homogenous quadratic polynomial. Modular forms are functions satisfying a certain form of equation and some other conditions. Conjunctive/disjunctive/Skolem normal forms are a particular class of first-order formulae with a certain syntax. Chomsky normal forms and Backus-Naur forms are descriptions of context-free grammars with certain syntactical restrictions on the production rules. And so on...
