Doctor Who has been wildly popular in Britain for 60 years: 26 doctors in 39 seasons (1963-present), plus spin-offs, over 200 novels, and enough tie-in products to rival Star Trek in the U.S.
The premise: The Doctor is a Time Lord, able to zap through time and space on his Tardis vehicle (which looks like a 1960s British police box from the outside). He has a tragic back story which might be new to this series: he is the only surviving member of his species. They were all wiped out by the evil ("Exterminate!") Daleks, but he destroyed their species in retaliation (until they return).
The Doctor is in the habit of saying "It's hopeless! There's no escape! There's nothing I can do -- we're all going to die!" Or "the universe will collapse at any moment! There's no way to stop it!" Or 'we're stuck forever on this parallel world where Britain has a president instead of a prime minister, and they've invented helicopters but not airplanes!" Then, after the commercial break: "I've figured it out! All we have to do is recalibrate the time coordinator and push it backwards through the space-time continnum!"
I'm reminded of the old Star Trek series, where Captain Kirk says "The odds against us getting out of this jam are a million to one!" Then he does it easily, and starts deciding what to wear for his promotion to Admiral.
The companion: In the first episode, the Doctor meets Rose Tyler, a working-class shop girl from 21st century London, and invites her to join him. Rose has a tragic back story, too: her father was killed in a traffic accident while she was a baby. Somehow the Doctor's missions often put them in parallel worlds where he's still alive (but she can't see him, or time/space will collapse), or back in time to the moment of the accident (but she can't rescue him, or flying gargoyles will destroy the world).
I don't know if the Doctor fell in love with his previous female companions, or this is a new innovation, but he and Rose are definitely falling in love. It's a slow burn romance -- we're halfway through Season 2, and they haven't kissed yet. Of course, Rose has a boyfriend, and the Doctor is busy falling in love with the lady alien or distant-future babe of the week (even Madame de Pompadour, when he tries to prevent distant-future cyborgs from stealing her brain).
Occasionally they pick up a second companion, a guy, but the Doctor resents the competition and quickly boots him.
The Guys: While they are in 21st century Utah, investigating an underground museum of alien artifacts, they pick up "boy genius" Adam Mitchell (Bruno Langley). He is fired in the next episode, when the Doctor catches him transmitting technology from the year 200,000 to his Mum's answering machine back home. Langley also played Todd Grimshaw, the first gay character on the long-running soap Coronation Street, from 2001 to 2003. He is heterosexual in real life.
Next, the Doctor and Rose end up in blitz-besieged World War II London, where alien technology has transformed a dead boy into an "empty boy," wandering around and asking "Are you my Mummy?" If he touches you, you turn into an "empty boy," too. During this adventure, they hook up with Captain Jack Harkness (John Barrowman, left and below), a loveable rogue time-traveler, and openly bisexual, flirting with men and women. Rose is shocked by this -- apparently LGBT people do not exist in 21st century London -- but the Doctor points out that Jack is from the 51st century, when "anything goes."