: What's the difference between a 1-, 2- or 3-dimensional character? The terms are bandied around quite often to describe fictional characters, generally in order to help describe how developed a

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I would firstly like to say that the answer by @Jay is excellent, and provides some good pointers on which characters should be one-dimensional or three-dimensional.
Like others here, I have never heard of two- or three-dimensional characters. I have heard of one-dimensional and multi-dimensional.
One-dimensional
This reflects a character that lacks depth, as if he was made of only one dimension. As Jay has pointed out, a one-dimensional character can often be summed up in one line or phrase. There's nothing wrong with one-dimensional characters, as long as they don't have a large part to play in the story.
Multi-dimensional
Multi-dimensional characters have several layers, facets, or dimensions to them. They are more complex and harder to figure out than one-dimensional characters. Once again, as Jay pointed out, multi-dimensional characters can take several lines to sum up, because they have several different characteristics. I believe it goes deeper than that though.

Every protagonist in a book (who should always be multi-dimensional) needs two basic parts: strength and inner conflict.
Strength
A lot of people like to focus on making sure their characters are flawed. This is fine, but what they seem to forget is that we have to like the characters too. Who wants to read about someone we don't care about?
Main characters need a strength, some quality about them that the reader can root for. Honesty, integrity, humility, humor, thrift, etc. The list goes on and on. The strongest of strengths are self-sacrifice and forgiveness (notice: they are both selfless qualities).
Inner-Conflict
Your hero needs strength, but with nothing else, he will still bore the reader. As @Filip has pointed out with Harry Potter, this is because they need something more: Inner Conflict.
Inner Conflict can manifest in two ways: either two opposing desires in direct opposition to each other, or sides of the protagonist (dimensions) that are incompatible, and therefore conflict.
An example:

During a war, a woman seeks vengeance against the enemy for the murder of her parents. At the same time, her sister is sickened by the war. Out of love for her sister, the woman seeks to end the war entirely.
The woman both wants to be a part of the war, and end the war, simultaneously. Her Inner Conflict pulls her in two opposite directions, both unrelenting.

Back to your question. Making a character multi-dimensional is not an easy process. One-dimensional characters often have a single purpose, determined by the plot, and a physical appearance. Multi-dimensional characters have a physical appearance, ways that they act, ways that they interact with others, and why they do what they do. If your character is a main character (but not the antagonist) they will also have inner-conflict; if he is a protagonist, he will also have strength and inner-conflict.
To form characters, you need to start with your protagonist.
Determine what makes your protagonist who he is. What defines him? What strengths are central to his very being? What inner conflicts keep him awake at night? Defining these two things alone will highly develop your protagonist. Then get in his head, figure out how he thinks, and from there, how he acts and interacts.
Once you have your protagonist down, all other characters are secondary. No matter how large of a part they play, they have a reason to exist, a reason that is linked to the protagonist. Maybe one helps him realize something he otherwise wouldn't. One is holding him back. One is pushing him forwards. One spreads doubt. One fans the inner conflict. One simply acts as a target for the protagonist's strength. One may see things the protagonist can't. (This is a different PoV character, and often acts as a secondary protagonist. This character needs strength as well.) The reason that you need side characters will determine who they are. You can then assign them inner conflicts (not necessarily strengths, unless they are very large characters in the story), and get inside their heads.
The further away you get from the protagonist, the less you will have to develop characters. Eventually you won't need inner-conflict anymore, and soon you won't even need to get inside a character's head and figure out who they are. You've arrived at one-dimensional characters, those people that pop into the story, perform their purpose, and then are never seen again.

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Some subjective examples in accordance with the definitions below:

1-dimensional: Katniss Everdeen. Harry Potter.
2-dimensional: I'm at a loss here, although I very much like Lauren's definition. However, I feel that remembering this special type of character is rather hard, since they neither annoy you like the 1-dimensional characters, nor do they stay with you like the 3-dimensional characters. They're just ... there, somehow. I'm pretty sure that you can find them in any major Blockbuster or Bestseller.
3-dimensional: William Stoner, Macbeth and Lady Macbeth, Davy Jones of Pirates of the Caribbean, Counselor Sam Healy of Orange is the new Black, and pretty much the entire ensemble of American Hustle.

Regarding the definition, I can offer a slightly different perspective that is however consistent with the definitions above. For me, 1- and 2-dimensional characters are soulless. I can just dump them into a story and they will serve their purpose, no questions asked. They feel hollow and uncomfortable when using them, like a piece of clothing that doesn't quite fit. 3-dimensional characters on the other hand are stubborn. They have their own mind and will flat out refuse to engage in any action that is in disagreement with their nature. It is still possible to force them into certain situations and actions, but writing these scenes will feel like a lie to you, until you surrender to your character and let him or her have her/his natural way.

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In the novel Flatland, a book where the characters are all geometrical figures, there are one-dimensional characters: lines, two-dimentionsal characters: squares and circles, and three-dimensional characters: cubes and spheres. :-)

Outside of that, we pretty much talk about "one-dimensional" versus "three-dimensional". The latter is also called "well rounded" or "well developed". I don't recall ever hearing someone talk about a "two-dimensional character".

By "one dimensional" we mean a character whose entire personality can be summed up in a sentence or two. He is a stereotype. If I tell you the character is a "rebellious teenager", a "greedy businessman", a "zealous political activist", etc, I've told you all there is to know about him.

A "well developed" character is much more difficult to describe. You might start out by saying that he is, say, a greedy businessman. But to understand the character, you would also have to explain that he is a loving father, a thoughtful Libertarian, an enthusiastic but unsuccessful baseball player, a lover of Mexican food, etc.

Think of real people you know. Could you describe them in one sentence? Probably not. Real people have not just many attributes to their personality, but often seemingly contradictory attributes. Someone may be excitable and emotional in some ways but very calm and reserved about other things. He may be generous with friends and familiy but tight-fisted with strangers. Hard-working at her job but lazy about housekeeping. Etc.

In general, the main characters in a story should not be one-dimensional. For the reader to be interested in them, they should be complex, realistic people.

Most stories have peripheral characters who can, and arguably should be, one-dimensional. For example, if there's a brief mention of the hero stopping at a store to buy whatever, the clerk will normally be a very simple, short-lived, one-dimensional character. He may be helpful or rude, capable or incompetent, as fits the story. But we don't normally expect a long discussion of his family background, his hopes and dreams, his religious and political affiliation, etc. In most cases that would bog the story down with irrelevant detail that would distract the reader from the main action.

But I've seen plenty of movies and read plenty of books where I found myself saying, "Yeah, yeah, I get it, this character is the enthusiastic but inexperienced young man, this one is the pretty girl who turns out to be tougher than any of the men, this one is the corrupt politician, etc." That makes them all boring.

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