Alghero's Catalan heritage shows up on the menu. Lobster (aragosta) is the signature dish — usually served alla catalana, split and grilled with tomato and onion. It's expensive but worth ordering once. The old town has a concentration of seafood restaurants along and around the port; quality varies, so walk around before sitting down.
The covered market on Via Cagliari has good fresh fish, cheese, and vegetables if you're self-catering.