At Cambridge and I think Oxford too, colleges are responsible for the non-teaching stuff. They'll do the admissions and interviews, rent you accommodation, offer personal support if you need it. You can get your mail sent to the college and pick it up at the pigeonholes at the entrance. The colleges usually have their own dining hall, common room, bar, chapel, and sports teams for the popular sports (rowing, football, rugby, etc). They have their own student unions (JCR for undergrads and MCR for grad students) who'll organise events. The college also sorts out your tutorials - a professor, you, and maybe one or two other students discussing some work the professor set.
The facilities are normally all at the 'central site'; most colleges also have accommodation on other sites and that's more like a 'regular' hall of residence or shared house. They like to put first years at central site so it's easier and you're less likely to feel isolated.
The university with its departments, apart from its colleges, does the teaching. So students from all colleges in each subject will be attending the same lectures and practicals, and sitting the same exams. There's also social activity, sports, and hobbies at university-wide level. If you want to do something less common like fencing, rock climbing, or ham radio, that will probably be done by a university-wide student group.
Durham I think is somewhat similar, but maybe the colleges do a bit less there and the university does more?
Trinity College Dublin is an unusual case. In theory it has a collegiate structure like Oxford and Cambridge, but there's only one college. So I think it ends up being a legal distinction with no practical effect at Dublin.