this answer has many variations as the different suites. Getting some quotes from stackoverflow, we can resume like this:
"Unit Tests
Tests the smallest unit of functionality, typically a method/function. Unit tests should be focussed on one particular feature (e.g., calling the pop method when the stack is empty should throw an InvalidOperationException).
Functional Tests
Functional tests check a particular feature for correctness by comparing the results for a given input against the specification. Functional tests don't concern themselves with intermediate results or side-effects, just the result (they don't care that after doing x, object y has state z). They are written to test part of the specification such as, "calling function Square(x) with the argument of 2 returns 4".
Acceptance Tests
Acceptance testing seems to be split into two types:
Standard acceptance testing involves performing tests on the full system (e.g. using your web page via a web browser [Selenium Module]) to see whether the application's functionality satisfies the specification. E.g. "clicking a zoom icon should enlarge the document view by 25%." There is no real continuum of results, just a pass or fail outcome.
The advantage is that the tests are described in plain English and ensures the software, as a whole, is feature complete. The disadvantage is that you've moved another level up the testing pyramid. Acceptance tests touch mountains of code, so tracking down a failure can be tricky.
Also, in agile software development, user acceptance testing involves creating tests to mirror the user stories created by/for the software's customer during development. If the tests pass, it means the software should meet the customer's requirements and the stories can be considered complete. An acceptance test suite is basically an executable specification written in a domain specific language that describes the tests in the language used by the users of the system."