Two and Three Dimensions
The artist M.C. Escher liked to play with images. The prints "Drawing Hands" and "Reptiles" for instance are quite famous for that.
Notice that the sleeves in the "Drawing Hands" print are flat. The hands are drawn to show depth. The hands look quite solid. The sleeves are 2-dimensional and the hands are 3-dimensional.
This difference in dimension is also present in the Reptiles picture. The sketch book on the table is flat and so are the lizards drawn one the page. But something funny happens towards the bottom. One of the reptiles seems to be crawling out of the page. As he crawls out of the page, he becomes more solid. He is shown as a completely solid little animal as he crawls over the book, over the triangle and the small bucket at the top-left.
A 2-dimensional object is an object that is flat. Examples are the sleeves in Drawing Hands and the drawn lizards on the page of the book in Reptiles.
A 3-dimensional object is an object that is solid. Examples are the hands in Drawing Hands and for instance the flower pot, bottle and glass in Reptiles.
From 2 to 3 dimensions
If you have ever folded a box from cardboard, or flattened a pizza box, you have gone from 2 to 3 dimensions or back.
In the image above we see the X pentomino. It is made up of 5 squares. The figure on the left is flat and hence is 2-dimensional. The figure in the middle is folded and bent a bit, but it is still pretty much flat. What about the shape on the right? Mathematically it is still really 2 dimensional because it is made up of flat pieces.
Now suppose we fill the cube with sand (or water). Now it's a solid object and it is 3-dimensional!