First episode: A Scandal in Belgravia
Seen also in: The Sign of Three (cameo)
Show history: Irene Adler crosses paths with Sherlock when she blackmails the royal family with pictures she's taken of one of its members. As Irene, or, as she is professionally known, "The Woman," is a dominatrix, these pictures are of an extremely compromising nature and a huge potential scandal for the royal family. Mycroft sets Sherlock on the case to reclaim the pictures, which leads Sherlock to a surprising discovery: Irene is nearly as intelligent as he is, and a worthy opponent within their battle of wits.
What starts out as a clash of the minds turns into a clash of hearts, as Sherlock and Irene's relationship becomes complicated as they grow more and more fascinated by each other. Before Sherlock, Irene has never met a man (or woman) who could resist her sexual charms; before Irene, Sherlock has never met a woman who is a match for his intellectual prowess. The potential for an intriguing, incredibly complex dynamic between them is crushed, unfortunately, when Irene is found dead, sending Sherlock into a quiet, but genuine, grieving period.
Sherlock's world is turned on its head, though, when six months later Irene reveals herself to be alive and well. She was in danger and had to disappear, and so left her camera phone with all her blackmail on it in Sherlock's care so that it would be kept safe until she could return. She asks Sherlock to decipher a coded email for her, which he does, and secretly sends Moriarty the decoded message. The message essentially brings Mycroft's operation for MI6 to a grinding halt and gives Irene the leverage to blackmail the British government into giving her whatever wealth and power she wants.
It looks as if Irene has won, until Sherlock realizes the code that unlocks her phone, which has read I AM _ _ _ _ LOCKED: I AM S H E R LOCKED. This reveals her genuine feelings for Sherlock, however ruthless her deception may have been, and unfortunately gives Mycroft access to all her blackmail, leaving her utterly defeated in all her strategic manipulations.
A few months later, Mycroft informs John that Irene has been executed while on the run in the Middle East. Though John tries to hide the truth from Sherlock, Sherlock hides his own secret from John: that Sherlock saved Irene's life just as she was about to be executed.
My take on her: I think Irene is a pretty polarizing character for the fans, for obvious reasons. Steven Moffat takes a classic character from the Sherlock Holmes canon and reworks her in a very clever, very modern, and very risky (and risque) way.
On the one hand, it's refreshing to see a female character who is very confident and comfortable with her sexuality (as well as being the first lesbian character on the show). On the other, that also makes her fall into the category of a cliched "strong woman" in Steven Moffat's repertoire: sexually voracious, good with a gun, etc. These characteristics, while not necessarily bad, are not what make a woman a strong character.
I am glad that they made her very intelligent; her scenes of going back and forth with Sherlock are incredible to watch. But in the end, she is also defeated by something that has become part and parcel of a stereotype for women in television and films: she is defeated by her emotions.
To reiterate: Irene can't win the game against Mycroft and the British government because she can't handle her emotions over Sherlock. I'm sorry, but that's just lame.
It's not completely one-sided, I suppose; Sherlock's emotions get the better of him, too, as he risks his life to save hers for no apparent logical reason - the only reason he could do so would be because of his affections towards her, however hard he tries to hide them.
So, while I can't hate Irene, I can't completely love her, either. She has such a fascinating build-up for her character, only to be let down by her inability to control her emotions - a disappointing resolution to what is otherwise a fantastic episode and an intriguing character.
Given her cameo in Sherlock's mind palace in "The Sign of Three," it looks like we may be seeing more of Irene in Season 4. Hopefully she can be redeemed from her cliched ending in "Scandal," and given a resolution more worthy of such an awesome character.