Two-dimensional things are flat — they can be measured in length and width, but they have no depth. Geometrical shapes like squares, circles, and polygons are all two-dimensional.
A sheet of paper may seem to be two-dimensional, but because it does have a measurable (if tiny) depth, it's actually three-dimensional. Your dog is also three-dimensional, but a picture of him on your computer screen is two-dimensional. You can also use this mathematical adjective to mean "superficial," or "shallow." A bad guy in a movie might be described as a two-dimensional character, for example, since he seems to have "no depth."