Ouch bhobba! Surely we can dumb that down a bit more?
In one sentence: A wave function is just a particular kind of mathematical function.
In the context of a real simple quantum mechanical system:
Imagine an electron in a box. Using quantum mechanics, you can't say exactly where the electron is in the box. The possible positions of the electron are given by the wave function for that electron. When you look for the electron, you find it in a particular position. The probability of finding it at that position is given by the wave function.
Move the electron to the double slit experiment:
The probability of finding the electron at any position on the screen is given by the wave function. The form of the wave function is determined by the set-up of the experiment (one slit or two...)