As a non-UK passport holder you require immigration permission that allows you to study in the UK. For most students with an offer for a postgraduate programme at LSE, the visa you require is a Student visa.
Please note that if you are currently in the UK with a Standard Visitor visa (also known as a tourist visa), a Short-term Visitor visa or with certain types of immigration categories, you cannot apply inside the UK for a Student visa.
If you are in the UK and you do not already have a Student visa, please contact the International Student Visa Advice Team to find out whether you are able to apply inside the UK.