The Doctor is now a lecturer of physics at St. Luke's University in Bristol. The University is a fictional university so no looking it up, it doesn't exist, the two universities in Bristol are Bristol and West of England.
One of those watching the lectures is Bill Potts who isn't a student but the dinner lady who normally serves the undergraduates chips at the canteen. Bill and The Doctor build up a rapport and Bill visits the Doctor throughout the year. Bill buys the Doctor a Christmas present but there is nothing recipricol. During the episode, there is snow on the ground which one way of implying the episode is supposed to be over a year.
Bill takes a particular interest in a undergraduate who we learn is called Heather later. Bill tries to befriend Heather but it doesn't go to well at the start, nerves gets the better of her. Later on, Bill manages to get with Heather and form a bond with her.
The Doctor soon realises something is going wrong and discovers that Heather is not human after all. Heather has been replaced by a morphing creature that drips water. The same creature appeared in The Waters of Mars. The Doctor takes Bill in the Tardis first to an underground cellar where the water creature follows then he visits an alien world and the alien turns up too. Finally, the Doctor takes Bill to a battle between the Daleks and an unknown other force. The Water Creature follows the TARDIS all the way.
The Doctor realises that the creature is following because of a promise that Bill had made to Heather when Heather was human. The Doctor tells Bill to release Heather from the bond which Bill agrees to and the Water Creature stops following Bill.