All residents in Bristol have to pay council tax to Bristol City Council. Full-time registered students do not have to pay this tax unless they live with someone who is not a student. When you have found your accommodation in Bristol you are legally obliged to inform Bristol City Council you have moved in. You will also need to contact them if you move while you are in Bristol. Full details on how council tax charges are calculated, how you can pay and whether you are eligible for any exemptions are available on the Bristol City Council's Council tax web pages.
If you move to South Gloucestershire details about how they calculate, collect and accept payment of council tax are available on their website.