The kosher rules are complicated, and as noted are interpreted differently by different people, but fish doesn't count as meat for those purposes: it can be served with meat or with milk. (The category is called "pareve" and also includes fruit, vegetables, grain, and eggs.)
Now, to complicate the explanation: some fish aren't kosher at all (I think it has something to do with the scales), and the things that we call "shellfish," such as shrimp, lobster, and crabs, are neither fish nor kosher. But if a fish is kosher, the kosher rules allow it to be served with either milk or meat.
(Note: I am not remotely observant, but I've had relatives who kept kosher, and my girlfriend follows the rules to do with seafood, so I know a little about it.)