A thin client PC is a computer that relies on another computer (server) to run its programs. The server is a faster computer with more memory (more expensive) that can run applications faster and store a lot of information. The thin client is a only responsible for displaying and reading data to/from the user. It can be slower and have less memory (less expensive). Multiple users can run applications and store data on the same server.
A smart client is pretty much the opposite, where the users computer has more memory and is faster so it can run the application locally. With this option the the PC is more expensive, but a server is not needed.