So how does one actually go about creating a character? There are usually two parts, a system mechanic and an act of imagination. The mechanic is something like a series of dice rolls or the distribution of points (As described in Character generation), and is part of the rules of the specific system, which do not concern us here. The act of imagination, however, applies to all games, no matter what the system. The player must imagine a concept for the character that will define the way that the character acts, and the way that they will play the character. Choosing this personality can be done in a number of ways.
Archetypes
The easiest, and often the most fun, thing to do is to choose a well known archetype. The muscle-bound and oiled barbarian, the cold and heartless Mage, the doomed albino elf, or the sly rat-like thief. Everyone knows just how such a character is going to act, and the player gets all the fun of playing it to the hilt ("Sorry boss, I think I just killed the guy you were waiting for." "Ignorant oaf, I don't know why I bother with you!"), while being free to concentrate on the plot of the game.
Antitypes
Alternatively you can turn the archetypes on their heads and create an antitype, something just the opposite of what might be expected. You might play a moody, poetry loving barbarian, a fun-loving and hard-drinking wizard, a shining Paladin with a corrupt heart full of secrets, or a matter-of fact thief who spends a lot of time in libraries. The other players will be able to see where you are coming form, the elements that you are using, but both you and they get the enjoyment of creating something quite different and unusual.
Of course even an antitype is still a kind of type, defined by a few very noticeable traits, and thus not entirely rounded out.
Keywords and Caricatures
In fact the most type-like of all kinds of characters are those created around a couple of keywords or defining points, a love of fire, a ruined leg, a funny nose, or a silly voice. Some systems encourage such characters, having lists of personality-types, moods or demeanours (even AD&D's alignments fit roughly into this category), from which you pick one or more to define your character. In other systems you choose the key-characteristics on the basis of dice rolls for stats. For instance if you roll up a human with a very poor physical strength then you might choose to make him a child, or a feckless aesthete, or a dwarf.
Whatever the reason such a character starts out with a few very noticeable characteristics, and usually gains more depth through the course of play. The caricature elements are not the whole of the character, but they are the starting point from which a better image can be built.
Fully rounded characters
The idea of rounding out a character from a framework is to some people the only way to make a character. When faced with the opportunity of designing a new character they will consider her past, the details of her parenting, her appearance, courses of study, early travels, employment, anything that might help to shape her personality. In play they will then draw on these things as explanations ("Alice won't talk to the man, he reminds her too much of the cruel miller in her village when she was little.")
Some GM's always have their players create a background before the game, or make use of flashback scenes to early childhood, or adolescence in the middle of play to flesh out the character. Some systems even have built in rules for generating the character's life step by step (such as Runequest and MERP), with tables for families, schooling, education, and employment. Generally, however, a player will prefer to decide these things for herself.
None of these methods is any better or worse than any others, nor do they cover all the possible variations of why you might design a particular character in a certain way, but the describe the major themes.