As mentioned by the top-voted answer, Martin Fowler discusses these distinctions in Mocks Aren't Stubs, and in particular the subheading The Difference Between Mocks and Stubs, so make sure to read that article.
Rather than focusing on how these things are different, I think it's more enlightening to focus on why these are distinct concepts. Each exists for a different purpose.
Fakes
A fake is an implementation that behaves "naturally", but is not "real". These are fuzzy concepts and so different people have different understandings of what makes things a fake.
One example of a fake is an in-memory database (e.g. using sqlite with the :memory:
store). You would never use this for production (since the data is not persisted), but it's perfectly adequate as a database to use in a testing environment. It's also much more lightweight than a "real" database.
As another example, perhaps you use some kind of object store (e.g. Amazon S3) in production, but in a test you can simply save objects to files on disk; then your "save to disk" implementation would be a fake. (Or you could even fake the "save to disk" operation by using an in-memory filesystem instead.)
As a third example, imagine an object that provides a cache API; an object that implements the correct interface but that simply performs no caching at all but always returns a cache miss would be a kind of fake.
The purpose of a fake is not to affect the behavior of the system under test, but rather to simplify the implementation of the test (by removing unnecessary or heavyweight dependencies).
Stubs
A stub is an implementation that behaves "unnaturally". It is preconfigured (usually by the test set-up) to respond to specific inputs with specific outputs.
The purpose of a stub is to get your system under test into a specific state. For example, if you are writing a test for some code that interacts with a REST API, you could stub out the REST API with an API that always returns a canned response, or that responds to an API request with a specific error. This way you could write tests that make assertions about how the system reacts to these states; for example, testing the response your users get if the API returns a 404 error.
A stub is usually implemented to only respond to the exact interactions you've told it to respond to. But the key feature that makes something a stub is its purpose: a stub is all about setting up your test case.
Mocks
A mock is similar to a stub, but with verification added in. The purpose of a mock is to make assertions about how your system under test interacted with the dependency.
For example, if you are writing a test for a system that uploads files to a website, you could build a mock that accepts a file and that you can use to assert that the uploaded file was correct. Or, on a smaller scale, it's common to use a mock of an object to verify that the system under test calls specific methods of the mocked object.
Mocks are tied to interaction testing, which is a specific testing methodology. People who prefer to test system state rather than system interactions will use mocks sparingly if at all.
Test doubles
Fakes, stubs, and mocks all belong to the category of test doubles. A test double is any object or system you use in a test instead of something else. Most automated software testing involves the use of test doubles of some kind or another. Some other kinds of test doubles include dummy values, spies, and I/O blackholes.