To curtail something is to slow it down, put restrictions on it, or stop it entirely. If I give up cake, I am curtailing my cake-eating.
Curtail is an official-sounding word for stopping or slowing things down. The police try to curtail crime — they want there to be less crime in the world. A company may want to curtail their employees' computer time, so they spend more time working and less time goofing around. Teachers try to curtail whispering and note-passing in class. When something is curtailed, it's either stopped entirely or stopped quite a bit — it's cut short.