Remote exploits are generally much more serious than local ones, but fortunately, remote exploits are much easier to prevent and are generally less common.
For example, suppose you're running bind (a DNS resolver) on port 53 of a publicly connected computer, and the particular version has a vulnerability whereby an attacker can send a badly formed query that causes bind to open up a shell that runs as root on a different port of the machine. An attacker can use this weakness to connect to and effectively "own" the computer.
This type of exploit is often called a buffer overrun or stack-smashing attack and is described in the article, Smashing the Stack for Fun and Profit by Aleph One (see http://www.insecure.org/stf/smashstack.txt). The simple solution to these problems is to make sure that you know which servers are listening on which ports, and that you're running the latest versions of the software. If a machine is publicly connected, don't run any more services than necessary on it.