What is the most general mathematical object that defines a solution to an ordinary differential equation? What is the most general object that defines a solution to an ordinary differential equation?  (I don't know enough to know if this question is specific enough.  I am hoping the answer will be something like "a function", "a continuous function", "a piecewise continuous function" ... or something like this.)

The object you are looking for is called a phase curve. For ODE $\frac{dy}{dx}=f(x)$ with $x$, we are looking for a solution of the form:
$$y=y(t), x=x(t)$$
The result is a graph on the $xy$ plane that pass the initial condition $y=y_{0}$, $x=x_{0}$. All of this are quite standard. The best literature on this is:
Differential Geometry: Manifolds, Curves, and Surfaces 
http://www.amazon.com/Differential-Geometry-Manifolds-Surfaces-Mathematics/dp/0387966269/ref=sr_1_8?ie=UTF8&qid=1299830752&sr=8-8 
And Ordinary Differential Equations, by V.I.Arnold. 
http://www.amazon.com/Ordinary-Differential-Equations-Universitext-Vladimir/dp/3540345639/ref=sr_1_4?s=books&ie=UTF8&qid=1299830826&sr=1-4
