808 refers to the TR-808 drum machine made by Roland. Popular use has shortened it to 808 and further redefined the term to mean a kick drum sound made using a sine wave and having one or more characteristics of the sounds created by a TR-808
An 808 snare, therefore, is a snare sound created by using either a hardware version of an 808 or a sample of said sound or by creating anew digital emulation version of the sound
fat snares on the other hand are a description of processing to create a "fat" sound usually meaning rich in mid-freq energy, possibly using tape saturation emulation, distortion and overdrive