A one-dimensional character is defined by one singular aspect of his personality or who exists solely to fulfill one particular, categorical role. [Think of the wise, old mentor who exists solely to teach the main character, be old and wise, and then killed off to give the main character motivation.]
A two-dimensional character has the facade of a real personality, but there is no depth or reason behind it. His goals, wants, and needs are defined, but there is no passion, logic, or believable motivation behind them. He is "flat." [Think of the main character whose clearly a Mary Sue, for whom you could list a bunch of adjectives and descriptors, but really, its quite boring, and whenever there's a plot point where this character must make a decision, you want to throw the book at the wall.]
And a three-dimensional character is, well, like a real-life person. [Think of your favorite character ever who you feel you know like an old friend.]
A four-dimensional character is like a three-dimensional character, but he can time travel. [Think of Dr. Manhattan. Or if you prefer, Dr. Who.]