Just a few thoughts, definitely incomplete:
While the developer is coding a task, the tester can be examining the specifications (or requests from the customer, if there are no formal specs) and writing the test plan. This can include a conceptual framework for what needs to be tested, but it should also include formally writing test suites (yes, in code) as well. This can be quite a challenge for teams moving to agile, as a lot of testers are hired without programming skills. (In a lot of places, it seems like it's a requirement to not be able to code.)
The tester can be involved in unit testing, or in a slightly higher scope by testing components or libraries that have a clean interface.
The testers should always be executing regression tests, load tests, and any other kinds of tests that he can think of, as well as writing test suites for the next sprint. It's often the case that testers work one sprint ahead of development (in preparing a test environment), as well as one sprint behind development (in testing what developers just produced).
