The word donburi refers to the actual bowl that the rice comes in. Sometimes this dish is called simply don or -don may be added as a suffix to indicate a donburi dish. For example, katsudon is short for tonkatsu donburi, which is "pork cutlet on rice." There are many other varieties such as gyuudon which is beef strips on rice, oyakodon (chicken and egg) and tempuradon which is tempura fried shrimp over rice.
Donburi is considered one of Japan's "fast foods" and there are chains of donburi restaurants specializing in quick meals for a low price.
Relevance to Urusei Yatsura
Apart from the fact that students eat it a lot on the show, Urusei Yatsura often does parodies of famous beefbowl franchise Yoshinoya (pictured above) such as a scene in the first UY movie.
There's also the occasional appearances of a masked wrestler who's dressed like a cow, and has dedicated his life to being a connoiseur of the beefbowl. On one occasion he beat the living snot out of Onsen-Mark because he offered him a bowl with extra toppings (the wrestler was offended by this implication that beefbowl wasn't already perfect on its own).