Back in the good old days, when you had to carry around a PCMCIA card in order use Wifi, and even before the term "Wifi" was coined, Wireless Access Points (WAPs or just APs) provided all the functionality of 802.11 wireless in a single device. Each AP minded it's own business and did it's own thing - communicating with clients over radio frequencies, encrypting and de-encrypting packets if necessary, and passing those packets onto the wired network. This was all fine and good when Wifi hotspots were - well, just that - "spots" - individual, isolated locations. But as companies and universities started deploying very large areas of contiguous coverage - sometimes with thousands of APs - some issues surfaced with this model.
For example, with individual autonomous APs, someone has to manually tune the power of radio signals on each AP so that, together, the APs cover an entire area. Also, as clients roamed between APs that had encryption enabled, the client needed to establish a new encryption key with each new AP which would take time and cause a short "outage" during the transition.
But, what if there was a central "controller" that controlled all the APs and knew everything all the APs knew ? Then the controller could tell each AP how strong it's radio signal needs to be in order to "fill" an area. And the encryption key could reside on the controller instead of the APs so that as clients roam between APs they don't need to renegotiate an encryption key.
Thus the terms "thick" or "fat" APs and "thin" APs were born ! With these new "controller-based" systems, the functionality of the traditional AP is split across the AP (or in HP speak Radio Ports) and one or more central controllers. This architecture provides a number of advantages in addition to the ones I mentioned and nearly all the enterprise-class wireless systems on the market today utilize this model.