A compressor balances the volume of the signal, not necessarily the strings (though often, then thicker strings are louder). What happens is that when you pluck a not, the sound has a certain amplitude (volume) and this can vary as you play across the strings and as a result of the different strings, pole piece heights, etc... What the compressor does is to make louder noises quieter and softer noises louder so that all sounds end up being roughly the same volume. This smoothes out the sound.
An overdrive is a completely different thing. This effect is meant to simulate the sound of an 'overdriven' tube (valve) amp circuit. What this means, roughly speaking, is that the tubes are pushed beyond their normal tolerances, which results in distortion of the sound wave. Depending on the components, this can result in a pleasing warmer, thicker sound for each note, with rich harmonic content. I'm sure some of the amp gurus on this forum can supply exact details about waves and amplitudes and scientific stuff, but all you need to know know is that an overdrive will thicken up and 'dirty' the guitar sound.