The tl;dr versions:
Story: A story is literally a story about what you would like your project to do: "I want to be able to find and read news", for example on HN. That's what they call items entered into pivotal.
Use case: One way someone might use your project to accomplish a task. "I would like to reply to an email" for example.
Acceptance test: A test, manual or automated, that is the criteria for whether something is acceptable for a given use case
deliver: There's a button, it says deliver, you click it. Usually means making something available for testing
deploy: Put your project onto a server where other people can use it
unit test: An automated test that tests a very small amount of code, a 'unit', and is used for making sure that unit works correctly. C.f. acceptance tests above, which test a much larger amount than one 'unit'. An example would be a test that tests one function, one method, or one line of code.