The other answers have explained pretty well what the advantages of a multicolor deck are, but I think I can add something on this point:
What goes into the process of deciding which colors and cards to combine? Do people decide to build a red-green deck and then find the best cards to use, or do they decide on the kind of cards they want and later figure that red-green would probably give the best results?
It depends on what level you play at. A beginner might very well decide to make a deck of a specific color combination, and then perhaps be guided toward a particular strategy by that choice of colors - for example, choosing to play red-green kind of pushes you toward playing a deck with strong creatures and burn spells, or blue-white would push you toward a control deck with counterspells, white removal, and a few flying creatures.
But at higher levels, people don't think that much about colors when building a deck, at least not right at the beginning - for one thing, even once you choose a color or colors, there are still many thousands of cards to pick from in those colors. Pro deck designers usually build around a specific card, or more often, a specific combination of two or three cards, which typically sets the color scheme for the deck. For example, the Exarch-Twin deck that's currently popular relies on Deceiver Exarch (blue) and Splinter Twin (red), which requires that the deck is going to be blue-red. Or Valakut decks need Primeval Titan, so they pretty much have to be green-red (unless you have access to Prismatic Omen, but that's a different story). The Tempered Steel deck from Scars of Mirrodin Block Constructed is an example of a deck whose color (white) was set by a single card - of course, that was only possible because the cards that Tempered Steel works well with happen to be colorless.
In some other cases (actually this is pretty common), you might start out with one or two cards and begin building a deck around those cards' color(s), but then you discover that the deck doesn't work very well without a certain kind of effect that you can only get in a different color. A good example of this is Pyromancer Ascension. You could, in principle, build a mono-red deck around that card, but in order to get the enchantment active, you typically need to find and cast two copies of two different cards - for example, two Lightning Bolts and two Shocks. That's not likely to happen before your opponent kills you, unless you have a way to filter through your deck looking for multiple copies of spells - and that's where blue spells like Preordain come in. This is why Ascension decks are blue-red, and in fact why a lot of different decks (like the Puresteel Paladin/equipment combo deck) like to add blue, even when you might not think it's necessary.