Cybermats are metallic rodent-sized, quasi-robotic creatures, possibly with a biological component, capable of detecting humans by their brainwaves. 

They have been programmed to leap upon their preys going for the throat for the purpose of injecting a neurotropic virus.  When the Cybermen plan to attack an isolated site, they sometimes send Cybermats on ahead, often going unnoticed as they slide across the floor,  seeming harmless, even cute until their true nature is discovered. 

The Second Doctor first  encountered them on the planet Telos where

they were used by the Cybermen to attack him and his Companions. 

The Doctor defeated them by using an electrified cable to create a magnetic field that disorientated and destroyed the creatures (MM). 

They were also used on the Wheel in Space, arriving in small white bubbles from the ship Silver Carrier.  Upon discovery aboard the Wheel, Bill Duggan gives them the affectionate name "Billy Bug" before learning that the Cybermats had corroded their way through the Wheel's supply of bernalium rods - a metal vital to the function of the ship's laser.

The Second Doctor destroyed them by broadcasting high voltage current phase contrasts (SS). 

By the time they were later used by Dr. Kellman to infect space beacon Nerva with the Cybermen’s Neurotrope X virus, the Cybermats appearance had changed, becoming larger and less cute looking and with a segmented carpace. (4D).