The terms are bandied around quite often to describe fictional characters, generally in order to help describe how developed a character is in terms of personality and backstory.
However, are there explicit cut-off points of when to describe such characters in this way? What makes a character go from being 1-dimensional to 2- or 3- dimensional? Are there any good examples in fiction to represent such characters?
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.
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.
For my example I'll use the character Johnny Lawrence from the movie The Karate Kid and the sequel TV sries Cobra Kai.
In The Karate Kid Johnny was a two dimensional character teetering on the edge of one dimensional by a single hair. For most of The Karate Kid Johnny was a stereotypical bully. He ruthlessly beat up on the main character Daniel Larusso and overly aggressive with his ex-girlfriend Ali Mills, destroying her personal property and forcing an unwanted kiss on her.
The one thing that kept Johnny from being purely one dimensional is this second dimension that peeked its head out a few times during the movie. In the first scene he appeared in, before his first encounter with Daniel and Ali, he was talking about his friends about how he wanted to use his senior year to "make it all work." Although the script was unclear about what it was he needed to make work it was clear that since he was referred to as an "ace degenerate" by his friends and that his goal was that he wanted to become an "ex-degenerate" that he was self-aware enough to know that he wasn't all that.
Late in the movie during the big karate tournament his sensei demanded he play dirty and "Sweep the leg." Johnny's reaction in that scene indicated he had a big problem with cheating though he was too cowed by his master to say any different or to defy him.
One more example of this second dimension is that after Johnny loses to Daniel he personally gives Daniel the first place trophy and says "You're okay, Larusso." He gave Daniel respect for his well-deserved victory and made it clear they were okay going forward.
Johnny received his much deserved third dimension in the spinoff series Cobra Kai in that we see the effects the decades have had on him. He's still got a hot temper and can be a real jerk at times, but we also see a compassionate side as he helps weaker, nerdier kids, like the kids he used to bully, become strong, confident young people. But that doesn't mean he's perfect, he's a deadbeat dad and a drunk. He has developed a resentment of Daniel Larusso in that Daniel has become a successful businessman, husband and father and he has not. But even though Johnny carries this resentment several times throughout the first three seasons we see Johnny and Daniel come within inches of becoming friends and the idea is suggested that the rivals are more alike than either cares to admit.
Frankly I can't offer a better example of a two and three dimensional character than Johnny Lawrence.
Some subjective examples in accordance with the definitions below:
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.
