Recently we have been adding automated tests to our existing java applications.
What we have
The majority of these tests are integration tes
Unit tests execute methods in a class to verify proper input/output without testing the class in the larger context of your application. You might use mocks to simulate dependent classes -- you're doing black box testing of the class as a stand alone entity. Unit tests should be runnable from a developer workstation without any external service or software requirements.
Integration tests will include other components of your application and third party software (your Oracle dev database, for example, or Selenium tests for a webapp). These tests might still be very fast and run as part of a continuous build, but because they inject additional dependencies they also risk injecting new bugs that cause problems for your code but are not caused by your code. Preferably, integration tests are also where you inject real/recorded data and assert that the application stack as a whole is behaving as expected given those inputs.
The question comes down to what kind of bugs you're looking to find and how quickly you hope to find them. Unit tests help to reduce the number of "simple" mistakes while integration tests help you ferret out architectural and integration issues, hopefully simulating the effects of Murphy's Law on your application as a whole.