(Some time passes wherein I pull out my hair.)
Do you know how hard it is to explain Doctor Who in one post? There are encyclopedias, literally encyclopedias, explaining Doctor Who.
There's too much to explain within the current temporal parameters. Let me sum up. Starting with The Doctor. (As always, I'll never spoil you for anything that matters without warning you.)
The Doctor is, naturally, the main character, our flawed yet much loved hero whose adventures through time and space form the basis of the show. "The Doctor" is his title, self-chosen, rather than his name, which is never revealed to us, even in the episode titled "The Name of the Doctor." For all we know, his name could be Bruce. Or something that sounds like dolphins crying in space ("So long and thanks for all the feels!").
Although he struggles with conflicts and challenges that are all too human, the Doctor himself is not human. He is a Time Lord from the planet Gallifrey. Are there other Time Lords from Gallifrey running around? Kind of, but to explain that would involve spoilers, sweetie.
What does a Time Lord do, you ask? He has a really big brain, capable of seeing all of time at once. And he's a kind of guardian, making sure nobody mucks about in the time stream, changing things. Except Time Lords, of course. But even then there are certain fixed points in time, things that HAVE to happen, that can't be changed.
Beyond that, Time Lords have a particular fondness for London and its home planet, saving us from complete and total destruction over and over again. Including one memorable episode where he prevented the Titanic from falling from the sky and crashing into Buckingham Palace, which would have really interrupted the Queen's tea. (He has a whole thing with the Queen, which you'll discover later, you lucky Anglophile.)
The Doctor is old. How old? Hella. <---SPOILER-ish. He's a Time Lord though and when Time Lords get mortally wounded they regenerate into a new form/actor. So he doesn't have to iron his face in the morning.
Each Doctor has his own unique outfit, some that should probably have landed on "What Not to Wear: Time and Space Edition." Celery? Enough said.
Is there a limit to the number of regenerations the doctor can have? That number was 12, but there were some....occurrences, shall we say, in the episode "The Time of the Doctor," which was the 800th episode. 800 episodes. The mind boggles. So could there be more than 12 doctors eventually? The BBC locked Moffat in a box until he fixed the problem the first time--rather brilliantly, I might add--and I'm sure they'd do it again in a double heart beat. Did I mention the Doctor has two hearts?
Here's a list of the Doctors, along with the actors who had the honor of playing him, as well as the years they were on. Doctors 1 through 7 are part of the classic series. The 7th Doctor also had a cameo in the Doctor Who movie produced in 1996, which featured the 8th Doctor. The 8th Doctor appears chiefly in the movie, some webisodes that led up to something special, and well, that something special.
The new series of Doctor Who began in 2005 with the 9th Doctor. The new series does not pick up where the classic series left off. What happened between the two series is alluded to, hinted at, and explained later in the new series.
One thing you'll soon discover is that everyone is passionate about "their" doctor. It might be the doctor from the first episode you saw, or the first one that you watched arrive on the scene through regeneration. But whoever it is, like the t-shirt says, you never forget your first doctor.
The 1st Doctor: William Hartnell (1963–66)
The 2nd Doctor: Patrick Troughton (1966–69)
The 3rd Doctor: Jon Pertwee (1970–74)
The 4th Doctor: Tom Baker (1974–81)
The 5th Doctor: Peter Davison (1981–84)
The 6th Doctor: Colin Baker (1984–86)
The 7th Doctor: Sylvester McCoy (1987–89, 1996)
The 8th Doctor: Paul McGann (1996, 2013)
The 9th Doctor: Christopher Eccleston (2005)
The 10th Doctor: David Tennant (2005–10)
The 11th Doctor: Matt Smith (2010–13)
The 12th Doctor: Peter Capaldi (2013–)
Anxious for more Who? Don't worry. We haven't even scratched the wibbly-wobbly, timey-wimey surface.