When looking for a PhD position what do we have to look for? How good the supervisor is or how good the institution/lab is?

Both!

Go to the best department with the best supervisor you can

What about a great famous supervisor in a not so great an institution..

What is the department itself like?

This department is one of the few places where a particular research is done and they have got more than 81 publications. The research team is really good and recently my masters supervisor told me it is the right department in a wrong university.. Also they offer full scholarship.

The department is more important than the Uni - there are some fab departments in some gawd awful institutions

Thank you all for clearing my worries...

In years to come, many people will ask who your supervisor was and who appraised your thesis. On that basis, it makes sense to try to get a supervisor who is well known in the field. So ditto earlier remarks - supervisor first, uni second.

Thank you silvester