The Shulhan Aruch (Yore De’a 173) codifies the prohibition against eating fish together with meat (listen to audio recording for precise citation). He writes that eating meat with fish poses the risk of "Dabar Aher," a euphemistic reference to a type of leprosy. The Shulhan Aruch emphasizes that since Halacha treats medical risks with greater seriousness than Halachic violations, one must exercise particular vigilance with regard to this prohibition. It is forbidden to eat a dish prepared with both fish and meat, and one should wash his hands before eating fish after meat, or vice versa. Furthermore, utensils used for fish or meat should not be used for the other unless they are first properly cleaned. Our custom is to treat chicken as meat with respect to this Halacha.
Many people have the practice to eat fish on Friday night before eating a meat dish. While this is certainly a laudable practice, one must ensure to separate between the fish and the meat as required by Halacha. The fish should be eaten separately from the meat, on a separate plate, and with separate cutlery. Furthermore, after eating the fish it is proper to wash one’s hands before eating meat.
Another context where these Halachot are especially relevant is weddings. The catering staff serving the food is, in all likelihood, unaware of the prohibition against eating meat with fish. If a person has meat on his plate and the person serving places fish on top of the meat, or vice versa, the food may not be eaten. One must ensure to have his meat and fish on separate plates, to eat them with separate cutlery, and to wash his hands in between. It is important to be cognizant of these Halachot and not to rely on the catering staff in this regard.
It is permissible for two people to eat fish and meat on the same table. The prohibition against eating fish with meat differs in this respect from the prohibition against eating meat with milk. When it comes to meat and milk, a person eating meat may not eat at the same table as somebody eating a dairy food unless they have an extraneous object on the table in between them. This restriction does not apply to meat and fish. It is thus entirely permissible to have meat and fish served to guests at one table at the same time.
Finally, the prohibition against eating fish with meat does not apply to cooking meat with fish utensils and vice versa. Assuming the pot or pan has been cleaned since it was last used for meat, one may use it for cooking fish, and vice versa, even within twenty-fours of the last use. The prohibition applies only to eating fish with meat; a meat taste imparted from a utensil into fish (or vice versa) does not render the food forbidden.
Summary: It is forbidden to eat meat (or poultry) together with fish. One should wash his hands after eating fish before eating meat (or vice versa), and clean or change the plate and cutlery. One must be especially cognizant of these laws at buffet affairs, and at the Friday night meal when it is customary to eat fish before meat. It is permissible to use a clean utensil for fish or meat even though it had been previously used for the other, even within the last twenty-four hours. It is permissible to eat fish at the same table as somebody eating meat.