A Ferrite Bead or Core is a hollowed out piece of Ferrite. Ferrite is a semi magnetic substance made from iron oxide mixed with other metals. They are used for a variety of applications.
Ferrite Beads and Cores help reduce RFI, Radio Frequency Interference. Most often they are used on cords that carry signals. (computers). Basically cords with signals traveling through them can start to radiate these signals like an antenna. The Ferrite Cores or Beads help reduce this signal radiation from these cords, that can cause interference to other electronics.
In the case of Ham Radio, Snap On Ferrite Cores are most often used to suppress a cords ability to act like a receiving antenna. For example, some of the Radio Frequency Energy coming out of an antenna can be picked by the microphone cord of the radio. This can cause the radio to squeal or howl. Or, in the worst case, it could burn your lip while using older gear. Snap On Ferrite Cores have been known to aid in the cure of some of these issues where you are "getting some RF back in to the shack".