A side, in a war, with air superiority, dominates the other side to the extent that it is implausible that the inferior side can put up any effective air missions. It is not, however, an impossibility, as it is in conditions of air supremacy.
An air superiority fighter is an aircraft designed for the sole purpose of attacking other aircraft, potentially deep in their own areas. To be a true air superiority fighter, it must not have additional roles, at least of ground attack, as illustrated by the United States Air Force saying about the basic F-15 Eagle series, "not a pound for air to ground".
Interceptors are also meant to fight other airplanes, but in specific kinds of engagements, such as point defense (e.g., the MiG-21 (NATO designation FISHBED), or a long-range fleet air defense interceptor (e.g., the F-14 Tomcat).