The definite integral from a to b of a function f(x) : [a,b]->R is the number

I = ∫ab f(x)dx

It is the area bounded between the x axis and the function f(x), and between the lines x=a and x=b (what "area" means in this context depends on what type of integral you're doing, as some "areas" may or may not exist according to your integral; in any case, when it exists, it matches your intuitions of what an "area" is).

When an indefinite integral of f(x) exists in the range [a,b], say the antiderivative F'(x)=f(x), we have that

ab f(x)dx = F(b)-F(a)
(this is the fundamental theorem of calculus, due to Newton and Leibniz).

However, many definite integrals "can be done" (i.e. may be expressed as an elementary function of "simple" constants such as 0,1,e and π) even when the corresponding indefinite integral has no solution as an elementary function of these constants and x. For instance,

-∞+∞ e-x2dx
has a known precise value, although the corresponding integral is an "impossible integral".