A teacher was a learned individual employed to teach others, typically in an institution, such as a school.
Two of the First Doctor's first companions were teachers. In November 1963, Ian Chesterton and Barbara Wright, who taught Susan Foreman, followed her to a junkyard as they were worried for her safety. They quickly became caught up in the Doctor's travels. (TV: An Unearthly Child)
The Second Doctor, Jamie McCrimmon, and Zoe Heriot, when visiting Dulkis, met Balan, a teacher and researcher of the residual radioactivity from a blast that happened 172 year prior. (TV: The Dominators)
Clara Oswald, companion of the Eleventh Doctor, became a teacher at Coal Hill School. (TV: The Day of the Doctor) At the school she dated Danny Pink, another teacher. (TV: Into the Dalek, Listen, The Caretaker)