Neil,

here's my explanation, in case it's of interest. (If it looks a bit too heavy, just jump to the ANALOGY at the bottom, which might be more helpful.)

A wavefunction is just an abstract mathematical function. Like y=x^2 or y=2x (which are both "functions of the variable x"). But in quantum mechanics the wavefunction is normally a complicated function of several variables (e.g. the position variables in 3 dimensions, x, y & z, plus time t). It's also a "complex" function, with real and imaginary numbers. (Don't worry if you don't understand what that means.)

A particle can have different wavefunctions depending on what physical attribute you're interested in / what it is you want to track and measure. You can have a wavefunction for its position, or momentum, or energy, or ...

At any instant in time, the wavefunction gives you a number for each point in space (for a position wavefunction) or for each combination of momentum in each direction (if it's a momentum wavefunction), etc. The number at that point in space and time is just an abstract number, but if you square that number you get the probability of finding the particle at that position (or momentum or ...) at that time.

And finally, the way the wavefunction spreads out over time and space is governed by the Schroedinger Wave Equation.

ANALOGY:

There's a really good analogy in the book "Quantum" by Jim Al Khalili which might help you understand what a wavefunction is. To paraphrase it slightly (to ensure I don't breach any copyright laws):

Imagine a burglar is out on a crime spree one night (a one-man crime wave, you might say). You know he's just burgled 101 Station Road a couple of minutes ago. But you don't know where he is now. So you get out a map of the town and assign probabilities to where he might be now. To start with, houses around Station Road are most at risk of being burgled next. But as time passes, the area he's likely to be in grows over time. (His "position wavefunction" spreads out.) And it might not spread out uniformly - perhaps he's more attracted to posh streets but less interested in houses around the police station.

This "crime wave", like any wavefunction, isn't something physically real, it's just a set of abstract numbers assigned to each position on the map.

And also, when you hear a report that he's just burgled another house, you have to re-do your figures (the wavefunction "collapses" to a small area around this new house) - but over time the wavefunction spreads out yet again.

Hope that helps.

Solvay.