Some History
The Mainframe Era
In the beginning of computing, there was the mainframe - one large machine which was fed batch jobs on paper tape or punched card. Users had to carry boxes of tapes or cards to the computer centre and wait for the results. This was helped by the advent of multi-user operation using attached dumb terminals, allowing many users to log on and run programs at the same time.
The PC ("fat client") Era
With the advent of the PC in the 1980s, the client-server paradigm was invented, whereby the mainframe was downgraded to a file storage device and the terminal was upgraded to a personal computer.
This was fine for a while, until large organizations started to realize that each of these PCs contained fans and hard discs which could break down, as well as a complex operating system which took substantial man-hours to maintain in good condition.
The Thin Client Era
In recent years, even low-end servers have attained previously unimagined levels of CPU power for ever-decreasing cost of ownership. At the same time, what was once the dumb terminal has developed USB ports, a sound card, a DVD drive and an ultra-thin LCD screen to become a fully fledged "thin client". That is, everything you expect from a PC but with no fan, no hard disc, no large-footprint steel casings and no complex, fragile operating system - and thus much cheaper to buy and maintain.
Along with this shift in economic realities, the drive to reduce software licensing costs has resulted in a move to replace software licensed per-seat with software which is licensed per-server.