Well, they're not all exactly the same - there are different exercises depending on the type of position and skills/competences required.
The invitation to the assessment centre usually sets out what the format will be, and whether or not there will be psychometric tests and/or personality questionnaires. Plus whether or not there will also be a one to one interview (although sometimes those are done on different dates).
Usually all the candidates are in one room with a number of observers around the edge. The number of observers will depend on the number of candidates and the complexity of the exercises. To start with, there's usually a short briefing, introducing everyone and a quick run through of what to expect.
Then there's a number of exercises during which the observers score the candidates according to the criteria that they have. The exercises are timed.
There's often an in-tray exercise which is designed to test a candidate's ability to take in information under time pressure, then prioritise and delegate where appropriate. The in-tray itself might consist of a wadge of letters/papers/memos and emails to sort through ... often with an urgent one 2/3 of the way down the pile. So the trick is to quickly scan each document sorting them into piles as you go 'urgent' important' 'not important' sort of thing. Then deciding on what takes priority and who is going to do it. You get the idea.
So ... a variety of exercises like that. Some are designed to test problem analysis and solving, some to test team work etc etc.