I'm going to come through this as simply as possible, as I tend to ramble, include strange comparisons to Smurfs and/or public transportation, and generally not make the most sense possible.
Sound waves are combinations of sinusoidal functions. The simplest sine and cosine pictures are here:
I'll be referring to the sine function (solid red line) exclusively from here on out; cosine (dotted blue line) is the same thing but shifted over a wee bit. The reason you hear stuff is because there is a variation in pressure in the air around your ears - the pressure increases, then decreases, then increases, and so on and so forth, at specific frequencies, and if you were to graph out the simplest possible pattern of increase and decrease it would look like that sine graph. 'Real' noises, though, are never exact sine waves, and instead are combinations of them. The combinations don't look like a single sine wave when you look at them in your DAW because there's a bloody lot of them, and they combine in silly ways.
The first thing to note is that the sine waves being combined are never very similar - lower sounds have lower frequencies (the 'humps' get fatter, as there are less of them filling a given period of time) and louder sounds have higher amplitudes (the 'humps' get taller), and there's just a ton of shit happening any time you have a 'real-world' sound. When it comes to actually combining the waves, it's really very simple - at any given point in time, you're basically adding the height of all of the different waves constituting your sound. Phase cancellation comes when you have two different waves being combined in such a way that one is high when the other is low, and vice versa. This happens everywhere naturally (when a guitar is out of tune, a 'beating' will be heard where the sound gets loud and then really soft at a given interval, when you listen to two speakers put in different places you'll hear some things adding and some things subtracting, the Fredman mic method enhances some low frequencies while killing some high frequencies, et cetera) and can be controlled very well for interesting effects; phase 'problems' come when a desired something-or-other is added or canceled out in a way that isn't desired (speakers wired backwards, microphones creating a 'comb-filtered' effect, waves bounce back from a wall into a microphone to make a guitar track sound 'boxy' or a room mic sound 'cheap') and that's about the end of that.
From a practical perspective, the only solution to phase issues is to move your microphones (when recording) or monitors (when playing back) if things seem to 'stack' in a funny way; you're not going to calculate the wavelength of the stuff that's going byebye, pull out the calipers, and figure out exactly how far things need to go, you're just going to move shit around until it's fine. With monitors, you can check polarity (as GuitarGodGT said earlier, one speaker 'pushes' while the other 'pulls' and you have a polarity problem), and with microphones you either move a mic or try to shift the track forward or backward a little so that it all adds up in a nicer way. Preventing phase problems is as simple as proper location (don't put amps against a wall or in a corner, or at a right angle to either, and be careful where you put drums and room mics) and carefully checking your sound when you use more than one mic - try monitoring drums in mono so you can tell early on if something is cancelling, don't use too many mics, et cetera. Hope that helped.
Jeff