If a carrier issues you a ticket for a flight operated by a different airline, that’s the result of a code-share agreement. This happens frequently between international alliance partners (SkyTeam; Oneworld; Star Alliance), but is not restricted to them. Be advised: not all code-share flights are equal in the eyes of your frequent-flier program; some may not count toward elite-qualifying miles or segments.
On the domestic side, small, regional carriers frequently operate short-haul flights on behalf of larger airlines. Sometimes they fly under their own names, but often they carry the brand of their domestic partner. The catch? The service, as well as on-time arrival and even safety records, of these regional airlines may not match those of the carrier with which you booked. The Department of Transportation is considering new rules that will make it easier to see who is operating your flight. Until then, look for the small “Operated by” notice when you search.