The most usual translation is “Shut up!” and if it is said (or shouted) on its own, it is pretty much the exact cultural equivalent of “Shut up!” However, the meaning is not identical.
The actual meaning of the word is usually given as “noisy” and that is very much the sense of the term, especially if we remember that “noise” is essentially unwanted sound.
We can describe noisy traffic as urusai; we can also describe a person who is too fussy as urusai. For example, we can even say that someone is “urusai about her clothes” – fussy about them. Again, the sense is that she makes too much “noise” over them.
So when urusai is used in the “shut up” sense, someone is essentially saying “Your words are unwanted sound”, thus “I don’t want to hear this”. Unlike “shut up”, urusai is not directly an order to stop talking, but a statement of one’s feelings about the talking.
Perhaps the second most common usage in anime (after the shouted “urusai!) is “urusai na”. The “na” in this case is a marker for a feeling expressed to oneself (though it may be “expressed to oneself” for someone else to hear). An English equivalent might be “what a noisy person!” – or possibly something less polite along the same lines.
The “feeling of sounds” is more important in Japanese than English, and that “sai” ending gives the feel of an excessive/unpleasant sensation, as in kusai, “smelly”, and extensions like mendokusai, “troublesome” (literally: “stinking of too-much-effort”).
Interestingly, while urusai is always negative, it isn’t necessarily critical. When I was in Japan recently we were in the direct path of a typhoon. We got torrential rain, as well as continual news reports and warnings about the incoming windstorm. For a while the television seemed to talk about nothing else.
However, at the last minute the typhoon changed course and veered northward toward Tokyo. The weather cleared up and went back to early-Fall sunshine and warmth.
My host mother commented:
“The television was urusai, wasn’t it?”
I don’t think she was criticizing the television at all. The typhoon was coming right at us and they would clearly have been failing in their duty if they hadn’t given warnings and news about it.
If the typhoon had hit us, she would certainly not have described the television as urusai. But as it happened it didn’t, and all that commotion turned out to be unwanted/unneeded “noise”.
In English she might have said “Well, that was a big false alarm, wasn’t it?” The use of urusai here represents the “false” (therefore unnecessary) part.
It was an interesting example of the fact that while urusai always means something negative, it doesn’t necessarily imply adverse criticism of the source of the “noise”.