A Rat King was a group of rats with interlinked tails.
On 9 June 1983, the Fifth Doctor encountered a Rat King in the Cadogan Tunnels. It was the leader of a group of sentient rats which had been experimented on by humans but had now taken control of them. The Rat King was killed by its creator, Dr Christopher Wallace, by its own punishment fluid. (AUDIO: Rat Trap)
The Seventh Doctor and Ace encountered a Rat King, which the Doctor called Rattenkoenig, on the planet Epajaenda. They also encountered a Rattenkaiser, or Rat Emperor. (PROSE: Last Rites)