totally agree with Drakkith. If you want another short description, here's my attempt:
The wavefunction 'contains' the physical information about a quantum system. Often it is a function of the positions of each of the particles in the system, and time. When there is just one particle in our system, we get the simple case. In this case, the squared magnitude of the wavefunction is equal to the probability of the particle being in some place, at some time.
Along a similar vein, for every physically observable quantity, there is a certain probability associated with each outcome (if we were to measure that quantity). And the wavefunction tells us what these probabilities are. Also, the form of the wavefunction of two identically prepared systems will be the same. (For example, two ground state hydrogen atoms).
Therefore, we can use the wavefunction to predict the average of many measurements of the same quantity in identically prepared systems. (this average is called an expectation value). Using the example of the ground state hydrogen atom, if we have a large number of ground state hydrogen atoms, and measure the radial position of the electron in each atom, the average will be 1.5 times the Bohr radius, although the radial position of the electron in a given atom will generally be measured to be something different to this.
Now, you're probably thinking that's all well and good, but how do we know what the wavefunction should be for a certain system? Well, we can use arguments about symmetries of the system, and the classical limit gives us an indication of the kind of laws the particles obey.
For example, the Schrodinger equation (which looks very similar to the classical equation of energy conservation), is often used to find the form of the wavefunction of a particle, given that we already know what the form of the potential energy of the particle should look like. Finally, the symbol psi is often used as the wavefunction. So if you see psi, then it is likely to represent a wavefunction, unless the author states otherwise.
