I usually categorize my tests in these 4 types:
1. Unit isolation tests – Test each unit in isolation. Written by Devs.
2. Unit integration tests – Test units in context. Written by Devs.
3. Integration tests – Test across the layers. Written by Devs and Testers.
4. Acceptance / Functional tests – Tests across the layers, but more focused on end user driving it. Written by Customers/Business Analyst and Testers, with the help of Devs. These are used for regression testing as well.