Why are we carbon-based?
We are carbon-based because carbon is the only molecule that is able to form the long, long chains of hundreds of molecules that are required to from proteins, DNA, etc. Absolutely no other molecule will do it. (Whatever you may have heard on Star Trek about Silicon-based life, Silicon doesn't form chains anywhere near as long.) Plus it's very handy in that its oxide is carbon dioxide, which is a gas, so using it and oxygen (another gas) as the end-points of respiration is very handy, you can get rid of a lot of gas very quickly. Silicon's oxide is, well, glass.

If you want to know why only carbon forms long chains, it's because of its size (not too large like silicon), its medium electronegativity (it's in a middle column of the periodic table, so it forms what are called "covalent bonds" very easily, especially with hydrogen, the #1 element in the universe by far; covalent bonds are easy to form and easy to break) and its valence (again, because of its position on the periodic table -- it forms four bonds, the maximum for a smallish molecule, giving it the geometry to form long chains).

For life to be based on anything but carbon, it would have to be made only of smaller molecules, which to our knowledge isn't very likely. Our DNA molecule is made up of billions of atoms; it's difficult to conceive how heredity could work on a small scale.