Can a broken/split chain be reassembled?
Fifteen years ago the answer was a simple "Yes". Today's 10, 11, and 12 speed chains are not designed to be reassembled this way, and in fact, will reliably break or fail where you reuse the rivet. For these new narrow chains, simply use a replacement rivet on Shimano, or master link on SRAM or other. 

For older bikes, or chains designed for 9 or fewer rear cogs, see below:

The same tool that you use to break the chain can be used to push the rivet back in. The trick is to make sure that you don't pop the rivet all the way out--it needs to stay in one of the sideplates. A lot of chain tools will stop automatically before they pop the rivet all the way out.

Alternatively, you can remove the rivet entirely, and the outer half-link, and replace that with a master link. You can get individual master links. These need to be sized to your chain (so if you have a 9-speed drivetrain, you'll have a 9-speed chain, and need a 9-speed master link).