Most of the functions we deal with in calculus are elementary functions. Intuitively, elementary functions are the ones you can write a nice formula for; the ones you know what to do with.
They're the power functions, logarithmic functions, exponential functions, trigonometric functions and their inverses, and all the functions you can build out of these by adding, multiplying, dividing, taking nth roots, and composing functions.
It might seem like we've just described every function there is. What could possibly be left? One example of a non-elementary function is
This is a function, but there's no way to write a nice formula for it using any combination of power, log, exponential, and trig functions.
Remember that there are more irrational than rational numbers? There are also more non-elementary functions than there are elementary functions. We don't encounter them as often in math classes because they're harder to think about. However, if you put all the functions that take real numbers as input and give real numbers as output up on a wall and threw a dart at them, you would probably hit a non-elementary function.