If it's elementary, then it can't be explicitly be expressed using standard mathematical arguments (powers, logarithms, trig functions, etc).
The simple integral of 2x^2 dx is obviously elementary. The answer uses numbers and powers.
The more complex and nasty integral of sqrt{1+x^2} dx is elementary as well. It has some inverse trig which makes it not fun to do for most, but it can still be expressed explicitly.
The integrals of e^(x^2), e^(-x^2), e^[(x^2)^2], and many others like these are NOT elementary because we cannot explicity state them with our normal mathematical lanuage, so to speak.
This is not a rigourous definition of elementary at all, but I hope it helps you understand the concept more.
To check this yourself, type in some of the integrals I gave you on the site ThePerfectHacker gave you a link to. The site won't be able to do some of the problems and on others it will give you an answer with something like, erf(x). All I'll say about the erf(x) is that it's NOT elementary.
Jameson