Many overseas students can do some work in the UK without needing a special work permit, for as long as they are registered as a student. Depending on your situation, there could be restrictions to the type and amount of work you can do.
If you are from a country within the European Economic Area (EEA) or a Swiss national, you can work in the UK without any restrictions on the type or amount of work you undertake.
If you’re from another country, you can check whether you can work or not by looking at the visa stamp in your passport. If you have a 'prohibited' stamp, you cannot work in the UK.
Otherwise, you should check the current rules on the UK Council for International Student Affairs website.
International students need to pay tax and National Insurance (NI) at the same rate as a UK citizen. This means that, in order to be employed in the UK, you will need to register for NI. Visit your local Jobcentre Plus to apply.