Whether you can work in the UK after your studies, and in what capacity, depends on the visa you apply for.
Tier 1 Graduate Entrepreneur (GE)
The Tier 1 Graduate Entrepreneur (GE) visa is for graduates who have a viable business idea that they want to put into practice in the UK. The University of Edinburgh is one of the endorsing institutions for this visa and can endorse up to 20 international graduates each year.
Tier 2 is the main immigration route for non-EU/EEA/Swiss nationals who wish to work in the UK. If you wish to stay in the UK to work after you have finished your studies there are concessions for Tier 4 students, which make it easier to get a Tier 2 visa.
Tier 4 Doctorate Extension Scheme
If you are studying on a PhD or other doctorate level qualification you can apply for a one-year visa to work or look for work in the UK.
Tier 5 (Temporary Workers)
The Tier 5 visa route allows non-EU/EEA/Swiss nationals to come to the UK to undertake specific types of work for a period of one or two years.