Each college is autonomous and self-governing, but each is also a member of the University. From a student's perspective, the college is their 'home' - it's where they live, eat, socialise (of course they'll make friends at other colleges too) and where their studies are arranged from. Many of their tutorials, especially in the early part of the degree, will be within their college. Each college has its own particular character and students usually become fiercely loyal to their own college.
The overall curriculum and assessment is set by the University and students will follow the same academic programme regardless of which college they are at (but the organisation of their course will be overseen by their director of studies in college). Lectures (and practicals for science students) are organised centrally, as are some classes. Exams are run by the University.
It doesn't really matter which college you end up at, because students nearly always decide very quickly that their college is the best and every other college is rubbish in comparison (including students who didn't originally apply to that college). On the other hand, since you get allowed to express a preference, you may as well (but if you really can't decide, you can make an open application and the computer assigns you to a college). There are certain practical things to consider - e.g would you rather be in a college with lots of beautiful old buildings or one with more modern accommodation? How much does self-catering provision matter? Would you like to be really central or to be a little further out and have more green space? Does it matter to you to have 3/4 years on the main college site, or would you be happy being in an annexe (probably with better catering facilities). Do you care if the library is open 24 hours a day? But lots of people pick a college for a totally random reason and there's nothing wrong with that. Noone will be asked at interview why they picked the college they did.
About a quarter of Oxford students end up getting an offer from a college they didn't apply to, so it's important not to get too hung up on the first choice college.
For open days, each college will be open to prospective students and their families, and so will all the academic departments. So you can visit a few colleges, and also go to the department(s) you're interested in. Oxford is a small city - you can easily wander round a good number of them on an open day.