The Gemara in Masechet Pesahim (76b) establishes that the consumption of meat together with fish poses the risk of Sara'at (leprosy). Accordingly, the Shulhan Aruch (Yore De'a 116:2; listen to audio recording for precise citation) rules that one must refrain from eating meat and fish together, in order to avoid their potentially harmful effects.
However, while Halacha clearly forbids partaking of meat with fish, this prohibition does not precisely resemble the prohibition against eating meat with milk. For example, the Shulhan Aruch (Yore De'a 88) rules that two acquaintances may not eat meat and milk at the same table. Even though one eats only meat and the other eats only dairy foods, they may nevertheless not eat together at the same table, given the concern that they may share with one another. This Halacha does not apply to acquaintances eating meat and fish; it is permissible to eat meat at a table where friends and acquaintances eat fish. (This often arises at barbeques and weddings, where people eat different foods at the same table.) This is the ruling of Hacham Ovadia Yosef, as recorded in Yalkut Yosef (Issur Ve'heter, vol. 3, p. 315).
The reason for this Halacha relates to the fact that eating meat with fish entails not simply a prohibition, but rather a health risk. We can assume that people will exercise more caution when it comes to their physical well-being, and the Sages therefore felt no need to enact safeguards to protect against mistakenly eating meat with fish. Such is not the case with regard to meat with milk, which is forbidden by force of the Torah, not due to health concerns.
Another difference involves the issue of "Beli'a" – absorption of taste. The Taz (commentary to the Shulhan Aruch by Rabbi David Halevi, 1586-1667) and most other authorities maintained that one may prepare fish in a utensil that had recently been used with meat, and vice versa. So long as the utensil is clean and does not contain any meat residue, it may be used with fish. Thus, for example, it is permissible to cook fish on a barbeque grill that was used for meat, provided that it is clean. So long as no actual substance of meat remains on the grill, it may be used for fish. In fact, one may even broil meat and fish together on different parts of the grill.
The exception to this rule is a closed grill, or an oven. When meat and fish are cooked together in a closed area, the steam produced by one is absorbed by the other. Therefore, one should avoid cooking meat and fish together in an oven or closed grill, unless the fish or the meat is covered, such as with aluminum foil and the like, so that it is shielded from the steam produced by the other. Nevertheless, after the fact, if one did cook fish and meat together in an oven without covering one of them, they are permissible for consumption.
Summary: It is forbidden to eat fish with meat. However, a person eating fish may eat at the same table with somebody eating meat. One should not cook fish and meat together in the same oven or closed grill, unless he covers one of them. Nevertheless, if one did cook meat and fish together in an oven without covering them, they may be eaten. One may cook fish on an open grill that had been used for meat (and vice versa), provided that the grill is clean, containing no residue of meat.