"Are you my mummy?"

Doctor Constantine: Before this war began, I was a father and a grandfather. Now I'm neither, but I'm still a doctor.
The Doctor: Yeah. I know the feeling.

This two-part episode is the first story Steven Moffat wrote for the revived Doctor Who. In what would become a continuing theme for Steven Moffat, it's very scary, contains tons of UST, and instantly won a Hugo award.

It introduces Captain Jack Harkness, who becomes a tremendously important character throughout the next few seasons and gets his own spinoff, Torchwood.

Colin Baker's favourite episode of all time.

The Doctor and Rose follow an alien spacecraft's emergency signal through time and space, to London during the Blitz. Since they can't very well leave alien stuff lying around in the past, they decide to retrieve it.

Things quickly get weird, though: the phone on the outside of the TARDIS rings, despite not being a real phone and not being hooked up to anything; a mysterious young woman appears and warns the Doctor not to answer it; when he does, it's a plaintive child's voice asking "Are you my mummy?"

Rose, in the fine old tradition of the Doctor's companions, has wandered off and gotten herself into trouble. In this case by grabbing a hold of a barrage balloon and accidentally taking to the skies above London while it's being bombed, wearing a t-shirt with a giant British flag on it. She's rescued in the nick of time by a suave spaceship pilot with an American accent, a RAF uniform and a penchant for flirting with his fellow soldiers, who introduces himself as Captain Jack Harkness. Captain Jack deduces that she's also a time traveler and, assuming that she's a Time Agent, tells her of a recently crashed alien ship, whose location he'll reveal for a few quid. Oh, and he (quite successfully) tries to seduce her — by taking her for a slow dance on the roof of his invisible ship, which is anchored to the top of Big Ben.

Elsewhere in London. Nancy, the young woman who warned the Doctor not to answer the phone, sneaks into a house whose inhabitants are in their bomb shelter. They've left the house mid-meal, so Nancy gives a signal and adorable little waifs come rushing in. They're mostly homeless kids and/or escapees from the Blitz Evacuation. Oh, and one extra person — adorable, all right, but unusually tall and deep-voiced, for a kid... it's the Doctor! He fishes for information but doesn't get very far. Then Nancy and the kids hear the approach of a certain kid in a certain gas mask, repeating a certain catch phrase, and flee in terror. Nancy identifies the boy as her brother, Jamie, and claims that he's "empty". The Doctor is left behind to see the kid up close — including a cut on the kid's hand — and is asked over and over, "Are you my mummy?"

Nancy, once tracked down, points the Doctor to "the Doctor". She means Doctor Constantine over at Albion Hospital, who treated Jamie when he first got hit by a mysterious falling object that wasn't a bomb.

Dr. Constantine is there, all right, along with a lot of apparently dead people wearing gas masks and bearing cuts on their hands, and they all got that way by touching Jamie or touching someone who'd touched him. And... they're still alive. Also, the gas masks are made of flesh and bone. As Dr. Constantine finishes mutating into a gas-mask-person, they all rise up from their beds and close in on our heroes...

Tropes:

Abandoned Hospital: Albion has all the trademarks of one, even though it wasn't really abandoned.
Blitz Evacuees: the kids Nancy takes care of, who are mostly runaways from the evacuation. The Doctor even invokes the trope.
Body Horror: The transformation of Dr Constantine, where you see the mask come out of his throat and hear his skull cracking.
Continuity Nod: Albion Hospital, the same hospital used in "Aliens of London". It has, however, actually moved in location during the last 60-odd years between the two adventures. According to the commentary, it migrated.
Cool Starship: Jack has a Chula Warship, which he tries to sell to Rose. He was planning to con her and actually sell a worthless ambulance, which would then be destroyed before she could get her hands on it, though.
Creepy Child: Jamie is even credited as "The Child".
Cymbal Banging Monkey: One is briefly seen as Nancy's whole house becomes possessed by Jamie.
Dead Little Brother: Nancy explains that Jamie was killed the day the mysterious bomb fell. This is what motivated her to look after the other homeless kids. It turns out Jamie was her son, but the trope still holds.
Distress Call
Eagleland: Riffs on this with the rivalry between the glamorous American womaniser with his expensive gadgets (Oversexed, overpaid and other here) vs. the Doctor's amateur 'British' approach.
Evil Phone: The Child can even animate the phone in the TARDIS's police box disguise, which isn't connected to anything.
Family Relationship Switcheroo: Former Trope Namer.
Glamorous Wartime Singer
Hell Is That Noise: We hear quite a few air raid sirens.
The Home Front: Food rationing and bombs feature prominently.
Noodle Incident: When the Doctor asks why the kids haven't left London yet, one of them mentions that "There was a man" where he was staying and gives no further explanation.
Not so Different: Constantine was a father and grandfather before the War, now he's neither, but he's still a Doctor. The Doctor quietly replies he's also this.
Pop Cultural Osmosis Failure: Rose gives the Doctor's name to Jack as Mr. Spock. Jack, being from far in the future and unfamiliar with Star Trek, calls the Doctor this.
Red Alert: Played with - according to the Doctor, mauve is the universally-recognised colour for danger, and most aliens consider red camp. "Oh, the misunderstandings, all those red alerts, all that dancing..."
Refuge in Audacity: The Doctor walks up to the mike at a bar during the Blitz and asks if anything's fallen from the sky recently. The audience seems to take the question as this trope and laughs.
Shout Out: Rose just won't give up on things being "very Spock". Guess what name she gives Jack for the Doctor?
Stealth Hi/Bye: Nancy when she first meets the Doctor.
- The Doctor ups this by walking past Nancy and the kids and somehow sitting at the head of the table, with no-one noticing his presence until he asks for a bit of dinner.
Street Urchin: Nancy and the other kids.
Title Drop: From Rose, to the show name:
Rose: Don't you ever get tired of "Doctor?" Doctor WHO?
The Doctor: Nine centuries in, I'm coping.
The Virus: Ironically, the nanogenes were trying to cure people as they didn't know what humans looked like and extrapolated it from the remains of a child with a gasmask on his face.
Wearing a Flag on Your Head: Rose, wearing a Union Jack t-shirt. She later notes (while hanging from a balloon) it's not the best idea during the Blitz.