Students do not avoid liability for council tax, unless they occupy a property equally with non-students. For example, you will be named on the bill (liable) if you hold a joint tenancy agreement with your housemates provided you are all students, as defined below.
However, students are 'not counted' (i.e. disregarded) in calculating the council tax bill, so a discount or full exemption may be due, depending on how many non-students also live with you.
If you are liable for the council tax bill, you are still responsible for ensuring any 'reduced' bill is paid.