An ordinary diff. eq. (ODE) is an equation of the form: f(x,y',y'',...y^(n)) = 0.
Of course, y has to be n times differentiable on at least an open interval.
The derivative of a function f is defined at a point, with the limit definition. If it exists, we call f differentiable at that point. If this holds for all points on an interval, we say f is differentiable on that interval. Since ODE's are 'built' with these derivatives, 'solving' a ODE is done, by definition, locally. However, we usually try to make the interval(s) I where the solution holds, as big as possible.
Example: y' cos²x = 1 is given for all x in R, yet the solution y = tan(x) is only valid in all open intervals (pi/2+k*pi,pi/2+(k+1)pi), k in Z of course.
In general, there's an infinite number of solutions to an ODE (of course, no solutions are possible as well!). Usually, we'll be interested in a particular solution, satisfying certain (initial and/or boundary) conditions. In this case, there are important theorems which give guarantees that locally, there exists a unique solution for a point in the plane; usually referred to as the "existence" and "uniqueness theorem". Note that these theorems gives a sufficient condition for a (unique) solution to exist, but they don't tell you how to find it! Similar theorems exist in other fields/applications as well.
Although it's possible that no solution can be given in "closed form", as a finite composition of elementary functions etc, it may still exist. Solutions are often given in implicit form as well, i.e. as f(x,y) = 0 rather than y = f(x). The implicit function theorem deals with these kind of functions, and how they can be locally unique, possibly solved to one of the variables under certain conditions.