I shall explain by example... I studied Computer Science at University College (Univ), Oxford. This meant that I was affiliated to Univ for all my residential \ pastoral matters (college wide), and affiliated to the Comlab and Maths Institute (university wide). Most of my tutors were Maths or Comp.Sci. professors or postgrads holding fellowships at Univ. Most of my tutorials and classes were held in Univ. Sometimes, however, classes were organised for students of more than one college held at one of those colleges; this would generally occur on an ad-hoc basis. 
All my LECTURES were given by Comlab/MInstitute professors to everyone studying that subject in the University. All exams were set centrally by the Comlab/Minstitute.
Not all colleges run all subjects. Some colleges are "better" than others for a subject, depending on how many they accept for that subject, and who they have teaching it.
I hope this helps!