The difference between The Unit, automated UI and Integration testing and when should we use each of them?

I have an ASP.NET MVC project which I've considered adding a test project on that.

My questions are about the following concepts:

• Unit Testing
• Integration Testing
• Automated UI Testing

I'd like to know the differences between them and when should I implement Automated UI testing rather than Unit or Integration Testing in the project?

Let's say that I've implemented Repository and Unit of Work design patterns. What kind of tests should I consider for testing them? Is it best practices to implement Unit Tests in order to test the database functionality or I'd better test it through Integration or Automated UI tests?

What type of test should I consider to test my controller's output? Integration or Automated UI Testing?