1. At least two female characters
2. Who talk to each other
3. About something other than a man.
All too often, when stories have a well developed, strong female character, if they have something "important" to discuss, they'll talk about it with a male character, while their conversations with other female characters will be limited to discussions about men. Meanwhile, male characters talk to other men about a wide variety of topics, which tends to imply that if a woman does something important, she's only able to do so by being "one of the boys".
Doctor Who, happily, has a long tradition of passing the Bechdel Test-- while not every episode passes, the very first episode, An Unearthly Child passes (Susan and Barbara talk to each other about Susan's schoolwork)-- and I think The Daleks and The Edge of Destruction pass too. From the second Doctor's era, The Invasion is a stellar example of an episode that passes significantly-- much of the plot development relies on conversations between Zoe and Isobel-- in particular, Zoe's computer skills and Isobel's photography.
I certainly don't have time to go through all of Old Who and applying the Bechdel test (although I suspect that eras in which the Doctor has more than one female companion tend to pass a lot-- as already mentioned, Susan and Barbara with the First Doctor, Nyssa and Tegan with the Fifth Doctor)-- although many of their conversations will be about the Doctor, and therefore they don't count. purplepen also reminds me that The Monster of Peladon is a good example of an Old Who ep that passes, when Sarah talks to the Queen about Women's Lib.
I also don't have time to go back and watch all of New Who at the moment, but since I have at least seen all of New Who, I'm going to start there. So I need your help-- where I don't remember enough of the episode, I'll indicate with a question mark, and if I miss something, please let me know.
I'm going to go a bit further than the Bechdel test usually does, and have three levels of "passing":
1. Low Pass-- Passes because of minor conversation/s that don't really drive the story in a significant way, although they still help add depth to the characters (and character development is a Good Thing), OR, the conversation/s only take place between minor characters who barely feature in the story as a whole.
2. Medium Pass-- The conversation/s pertain to a significant plot point, but doesn't play a major driving role in the narrative.
3. High Pass-- The conversations play a significant role in driving the narrative forward.
There's also the question of what "counts" as a conversation about men-- for instance, what if the two women are talking about a plot to blow up the earth-- and the plotters are men? While this isn't perfect, I'm going to say that it's a conversation about men/a man if they mention individual/s specficially (therefore it doesn't pass), but if they're discussing events or ideas, it passes. So, for instance, if they discuss the Battle of Waterloo, it passes; if they discuss Napoleon as an individual, it doesn't.
It is also okay if they discuss men at some point-- it's just that that can't be the only focus of the conversation-- for instance, Rose and Gwenyth talk about men in The Unquiet Dead, but they also discuss school and what Gwenyth sees in Rose's mind.
And for the purposes of discussing aliens, we'll assume that they're male if they're referred to by masculine pronouns-- I don't want this to descend into the discussion of the canonicity of the Tiemcock. ;P
Low Pass-- Jackie and Rose discuss Rose's work blowing up, and what Rose will do in the future. This is mostly irrelevant to how the plot develops (as Rose goes off with the Doctor), but it does establish their relationship and helps define both their characters.
1.02: The End of the World
1.03: The Unquiet Dead
Medium Pass-- Rose and Gwyneth discuss a variety of topics, including Gwyneth's psychic abilities, which are highly important to the plot development.
1.04, 1.05: Aliens of London/World War Three
Not sure-- As far as I can recall, Rose and Jackie mostly talk about the Doctor, and when Rose and Harriet Jones are together, the conversation is mostly centred on the Doctor-- I can't recall and significant sections of conversation where they're specifically talking to each other. Help?
Fail-- The female characters talk to men, not to each other.
1.07: The Long Game
1.08: Father's Day
Fail-- All conversations between Rose and Jackie are about Pete.
1.09, 1.10: The Empty Child/The Doctor Dances
Low Pass-- Rose and Nancy discuss the outcome of WWII.
1.11: Boom Town
Low Pass-- Margaret Slitheen and the female reporter discuss mysterious events surrounding the Blaidd Drwg power plant, and other details of their lives (some of which involves men, but not always).
1.12, 1.13: Bad Wolf/Parting of the Ways
Fail-- As far as I can recall, Lynda-with-a-y and Rose don't ever have a conversation that is just with each other, whenever women in the game shows talk to each other there's still a man involved in the conversation. I'd love to count the conversations between Jackie and Rose in PotW, but those (a)include Mickey, and (b)are framed around Pete.
2.0: The Christmas Invasion
Fail-- while there are significant conversations between female characters (Rose and Jackie, Rose and Harriet), they tend to focus on needing the Doctor back in action. I can't recall any snippets of conversation between Rose and Harriet that aren't about the Doctor, but please let me know if I'm forgetting something.
2.01: New Earth
Medium Pass (I think)-- Rose and Cassandra talk to each other, and I think Novice Haim talks to the cat-nun in charge of her. These conversations pertain to major narrative points.
2.02: Tooth and Claw
Medium/High Pass-- The women of the household band together to escape the werewolf and prepare mistletoe.
2.03: School Reunion
Low Pass-- A damn shame, because this one really should get a high pass-- unfortunately, Rose and Sarah Jane pretty much only talk about the Doctor. It gets a low pass, however, because they do discuss the sonic screwdriver at one point, and because Rose talks to the other chip ladies (who are really aliens, but, wev). Fortunately, every episode of SJA gets a High Pass. :)
2.04: The Girl in the Fireplace
2.05, 2.06: Rise of the Cybermen/Age of Steel
Low Pass-- Rose and alt!Jackie discuss how she likes her tea.
2.07: The Idiot's Lantern
2.08, 2.09: The Impossible Planet/The Satan Pit
(Very) Low Pass-- minor conversations between female technicians on the ship (h/t uncledark)
2.10: Love and Monsters
Fail-- while there is more than one female member of LINDA, they don't tend to talk to each other specifically; they talk to the whole group, or to male characters. Rose's desire to protect her mother is very sweet though.
2.11: Fear Her
Medium Pass-- Rose and the mother talk about the child and singing (h/t leighwoosey).
2.12, 2.13: Army of Ghosts/Doomsday
Fail (I think)-- Rose and Jackie only have conversations that include the Doctor, Yvonne only talks to men that I can remember, as does [Adeola?] Martha's cousin.
3.0: The Runaway Bride
Fail-- I can't recall any significant conversations between two women, and the little ones that I can recall all pertain to Lance.
3.01: Smith and Jones
and I think she talks to one of her female colleagues after the hospital arrives on the moon.
3.02: The Shakespeare Code
Low/Medium Pass-- All the Carrionites talk to each other, and they are central to the plot, BUT they're usually talking about Shakespeare or the Doctor.
3.04, 3.05: Daleks in Manhatten/Evolution of the Daleks
Medium/High Pass-- not much in the first ep, but later on Martha and Tallulah discuss Dalekanium and building plans, both of which are crucial to the story (h/t leighwoosey).
3.06: The Lazarus Experiment
Fail (I think)-- Martha talks to Tish and her mother, but as far as I can recall, all conversations are about the Doctor or Lazarus.
Low Pass-- Martha's phone calls to her mother.
3.08, 3.09: Human Nature/Family of Blood
Medium Pass-- Martha has significant conversations with both Jenny and Joan-- and while lots of them pertain to the Doctor/John Smith, not all of them do. Martha's relationship with Jenny, in particular, helps drive the plot, because we see the contrast in Jenny's behaviour when her body is overtaken by Mother of Mine, while her conversations with both women also contribute significantly to her character development.
Medium Pass-- Sally and Kathy are only together for a short space of time, but they do talk about things other than men, and their relationship is pretty crucial to the plot. Unfortunately, however, after Sally visit's Kathy's grave, I don't think the ep passes at all.
3.11: Utopia (Seperating these last three because, even though they are the same story, Martha interacts with different female characters in all three)
Low Pass-- Martha and Chantho talk about speech taboos (but mostly about how they're in love with the Doctor and Prof. Yana respectively).
3.12: The Sound of Drums
Low Pass-- Martha talks to her mother, but it's about a reconciliation with her father (not to mention being part of Saxon's evil plot). Vivian and Lucy talk to each other, but only about Saxon. This only gets a low pass becuase Vivian talks to Tish about Lucy-- it just scrapes in.
3.13: The Last of the Time Lords
Low Pass-- There is a pretty significant conversation between Martha and Professor Docherty, but Tom is involved in that. Martha does, however, present Prof. Docherty with flowers at the end to the story.
4.0: Voyage of the Damned
Fail (I think)-- Do Astrid and Foon talk to each other at any point?
4.01: Partners in Crime
4.02: Fires of Pompeii
Medium/High Pass-- Donna and Evelina talk, revealing an important plot point (the stone in her arm). The Sybiline sisters talk amongst themselves.
4.03: Planet of the Ood
Fail-- As far as I can recall, Donna never has a specific conversation with Salame (or whatever her name was).
4.04, 4.05: The Sontaran Stratagem/The Poison Sky
Fail-- A BIG shame, because potentially this one should be able to get a high pass, with both Donna and Martha in action. Unfortunately, they only really talk about the Doctor with each other. And Donna only talks to Sylvia when Wilf or the Doctor are involved.
4.06: The Doctor's Daughter
Low Pass-- Again, a big shame, particularly since we have THREE strong, major female characters here. But the only time I can recall two of them having a conversation that isn't about the Doctor is right at the beginning where Martha says to Donna, "I love this part," and Donna responds that she thought Martha wanted to go home. All three women do kickass things, but Martha is separated from the group, and Donna and Jenny tend to address the Doctor rather than each other. It is probably significant, however, that Donna gives Jenny her name, even though the Doctor is involved in that conversation.
4.07: The Unicorn and the Wasp
Fail-- Disappointing YET AGAIN, because there are lots of strong female characters here too-- but when Donna and Agatha speak privately, they only discuss Agatha's husband, Lance, and the Doctor, while the relationship between Lady Eddison and Miss Chandrakala isn't really developed on screen (please let me know, however, if there is in fact a conversation between the two of them at some point-- I just can't recall one).
4.08, 4.09: Silence in the Library/Forest of the Dead
Medium/High Pass-- Donna and River talk to each other, in a way that foreshadows Donna's larger character arc, and Donna and Miss Evangelina have significant conversations in both eps that pertain to both character development and driving the narrative forward.
Fail-- Unfortunate again; lots of good female characters, but none of them talk to each other-- they all just talk to the Doctor, or to the whole group.
4.11: Turn Left
High Pass-- In spite of this High Pass, it's worth noting that most conversations between Rose and Donna are about the Doctor-- but they do discuss important things that aren't the Doctor, and Donna also talks to Sylvia and her female friends at different points in the episode. And of course, Donna and the Fortune Teller talk to each other.
4.12, 4.13: The Stolen Earth/Journey's End
As I said above-- please help me out with the episodes that I don't remember, or if I've made any mistakes! At some stage I'd like to do Torchwood as well. There's no real point in doing SJA, however-- as I said above, every SJA episode gets a High Pass. Please feel free to talk about Old Who eps that pass the test in the comments too-- even though I don't have the time to go through them all, I'm very interested to hear about them, and discuss them. :)