In IT, a thin client is either a machine or a piece of software which is very light to install and configure and depends on an outside server who does all the donkey work.
As users are becoming more lazy, IT folks need to make things easier for them. That's why we shift all the hard dirty computational memory consuming work to a slave server or perhaps a cloud so users can work in a clean comfortable fast thin environment.
Think of a thin client like a fancy restaurant. Customers enjoy the atomosphere and service while the crew inside the restaurant work non stop to satisfy their needs. Customers do not care what's going on there. They request their food items and it get prepared, cooked, decorated and served.
Web is an ocean full of sexy thin clients, the browsers are the software and the servers to which the requests are sent is the fat clients.
Although this might seem to be a great architecture , each architecture must have some hidden flaws.