A one-dimensional character really has only one motivation/notable trait/shtick going for them. They might just be used for a simple gag the show keeps coming back to, or maybe they tend to have the same reaction to most things.
A two-dimensional character, rather straightforwardly, has a second motivation/notable trait/shtick. Like the literal difference between one and two dimensions, this actually adds a LOT, because the two aspects can interact and combine as part of the characterization. Two-dimensional characters are very common in most decent fiction.