I always used 3/4" A/C plywood when replacing or covering damaged subfloor. It costs a little more than 1/2", but when I considered the time and effort of re-doing the floors it seemed like a small price for a large improvement.
For those of you that don't know, the A/C refers to the quality of the two sides of the sheet of plywood. "A" is the best and means any knots or holes have been filled so the entire surface is smooth. This is especially important if you plan to put down vinyl flooring.
Vinyl flooring will quickly show any bumps, dips, or other defects in the surface of the subfloor. We used to joke that after walking on it for a few weeks you could call heads or tails on a dime left under it.
If you plan on covering your floor with carpet & good padding, or put down a wood laminate floor you should be OK using C/C plywood.
"C" grade means the wood is structurally sound, but will have assorted knots that have fallen out, cracks, and the surface will be rough. It's also much cheaper and none of that matters for the side that will be down.
I have been seeing reports that with demand down and profits under pressure some plywood sellers are selling B/C plywood as A/C. If the plywood you are buying is A/C it should be stamped that way. If you can't find that stamp on the sheets you are looking at, or the price seems really cheap, you might want to shop somewhere else.
For more information you might want to look at the Wikipedia article on plywood