It depends on the company. Functional testing and performance testing are common. Basically, you need to make sure your code works and integrates into the larger picture. Same deal with Scrum. You need to produce working code and regression tests.
Officially, Scrum is a process framework, not a methodology. There is actually nothing in Scrum that specifically calls out how testing should be done, however, technical practices like Test-Driven Development (TDD) and Continuous Integration are often adopted by Scrum teams. Many teams, agile or not, also adopt Behavior-Driven Development (BDD) and Acceptance Test-Driven Development (ATDD). There are many other different kinds of testing that teams will also consider doing such as Exploratory Testing, Load Testing, Security/Vulnerability Testing, Longevity Testing, to name a few more.