Isn't it obvious? Well, apparently not quite, when you consider how many people are still struggling with this topic!
The first concept you must grasp is that 3D means 3 dimensional and 2D means 2 dimensional. Now before you think I'm stating the obvious, let me go on to say that the 3D and 2D in animation refer to the dimension in which the animation was created. Ahhhh. The plot thickens eh?
For 2D animation, everything happens on a 2 dimensional platform. Pictures are flat, without depth and offer only one perspective. Objects and characters are usually drawn without the subtle soft shadows we see in real life and colours have few varying shades. In 3D animation, everything happens on a 3 dimensional platform. Pictures have depth and offer multiple perspectives just like in real life and have soft subtle shadows casted on the objects and characters within.
In 2D, characters look cartoonish and unrealisitc. In 3D, characters can look cartoonish but realistic at the same time.
Another way to think of this is to think in terms of a painting and a sculpture. 2D is a painting, and 3D is a sculpture. 3D introduces "depth perspective," so we not only see a rectangle (2D) but a CUBE (3D). You may also want to think of it like being the difference between a photograph of a glass of water (2D) and being able to reach out and actually pick up the glass of water (3D).
Typically, 2D involves "drawing," or movement on, say, a flat surface (sketch pad, etc.) or in the vertical and horizontal planes. 3D involves "modeling," i.e., creating objects in 3-dimensions using a computer software, residing in an expansive virtual environment, complete with lights, reflections, other objects, shadows, etc.
You could start training yourself by comparing a cartoon like Bugs Bunny, Aladdin, Lion King (2D) to "Toy Story 1,2 & 3, "Finding Nemo" and "Incredibles" (3D). If you have not watched any of these great cartoons, you should grab one right away or be branded a neantherdal forever!
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