The thin vs. thick distinction usually refers to how much processing or "business logic" is done on the client.
Basically, a thin client is a web based application and most of the processing is done on the server side.
Web browser is the classic example of thin client. It can be the client application for anyone's server application.
The client can be generic like a web browser, and since most of the logic takes places on the central server, it's much easier to push updates out to the clients.
Advantage of thin clients is they make fewer demands on the client machine, which can be anything from a computer to a smart phone to a household device like a blender or TV set top box.
A thick client is more like a standalone application, which run on the client machine and communicates with the server less frequently.
With thick client, there won't be much processing via the network. In a way, it will be a much faster option if your network is slow or congested.
A virus scanner is a good example. It downloads new virus definitions from the server, but then runs its scan on the client machine without further communication to the server.
Advantage of thick clients is that the performance isn't tied to the load on the server, and the speed of the network connection.