"The Lodger" is the eleventh episode of the fifth series of Doctor Who. It stars Matt Smith as The Eleventh Doctor, Karen Gillam as Amy Pond, and James Corden as new companion Craig Owens. Although Karen Gillam is listed as the co-star in the opening credits, this is actually a "companion lite" episode, with only a few scenes featuring Amy.
The TARDIS sets down in modern day England, and the Doctor exits. Before Amy can join him, some type of temporal phenomenon kicks in, whisking the TARDIS away, and leaving the Doctor stranded. The Doctor must get to the heart of the mystery and retrieve his TARDIS. He finds that the center of the mystery is located in a typical suburban home that luckily has a room to rent. The Doctor rents the room, becoming the eponymous lodger, and sets out solving the mystery.
The strength of the episode doesn't come from that storyline, as much as it comes from the comedy of The Doctor's lodging. For the man he is lodging with, Craig, is a portly young man, content to stay at home and muddle his way through a not-quite-relationship with girlfriend Sophie, played by Daisy Haggard. Not having a taste for the unusual, Craig is unsure about what to do with the eccentric and unusually flustered Doctor. The episode plays as a fish out of water/buddy comedy as The Doctor's antics help draw Craig and Sophie out of their shells, in the end getting them to admit their feelings for each other---and saving the solar system in the process.
There is nothing in this description that strongly recommends the episode: "fish out of water" is a very old schtick. What makes it work is that, more than most of the people who have played The Doctor, Matt Smith knows how to play him as an alien. And James Corden plays Craig as a timid suburbanite quite well. Together, the jokes and gags of the universe-spanning demigod doing things like fixing an omelette and playing football work perfectly.
Although seemingly a breather episode, it actually develops deeper issues, such as The Doctor's nomadic nature versus the domestic lives of the people he moves about. It also (perhaps) shows the first appearance of The Silence, villains that would become much more important very soon.