Very quick and dirty: RMS is a sort of average and peak is the top level. RMS corresponds better with what we hear as humans. We don't actually hear the peak level, we hear a sort of average power.
RMS means to take the root of the mean and square it. (EDIT: this is wrong! see post number 18 in this thread) This gives a sligthly different result than to take an average or mean in the usual way.
Not so easy, but hopefully useful too:
With a sine wave seen as a circle, the RMS is exactly at 45 degrees, halfway to the top.
This pic is from my website and shows the relation between the circle and a sinewave. At 45 degrees, the sine value is .707. This is also the amplitude value of the resultant waveform. .707 may seem like an odd number, but it's actually -3dB. It's also half the square root of two, which makes utterly sense if you look at it geometrically.
In comparison, a square wave have an RMS power of -6dB compared to the peak, noise is around -12dB. Music is naturally somewhere between -14 and -20. Until loudness mastering took off, that is. In these days, music resembles square waves more than music..