Atmel AVR® (or in short AVR) is a brand name of Atmel for several of their RISC based microcontrollers.
Several LinkStation types contain an Atmel AVR microcontroller, responsible for controlling the font-panel LEDs, the on-off button, the reset button and the watchdog timer. Other LinkStation types (at least the LS2) don't contain an Atmel AVR chip. Instead e.g. a Freescale microcontroller is used. Still, however, people also tend to call this microcontroller AVR, although it is technically wrong.
There is no general agreement about what 'AVR stands for. Atmel says it is just a name. Other speculations say it stands for Advanced Virtual Risc. Yet others claim the original enginners of the AVR series chips are named Alf Egil Bogen and Vegard Wollan, and AVR is short for AlfVegardRisc.