Harry discovers that it is Quirrell who is after the Stone and serving Voldemort, not Snape. Quirrell tries to use the Mirror of Erised to find the Stone but the Mirror gives it Harry. Voldemort realizes this and as part of Quirrell orders the professor to kill Harry, who blacks out during the struggle. He awakes in a hospital bed and hears many things from Dumbledore. At the end of term banquet, Gryffindor is awarded the House Cup.
Calendar and Dates
The action of this chapter picks up immediately after that of the preceding chapter, and continues through the remainder of term, ending as Harry, Ron, and Hermione leave Platform 9 and 3/4 at King's Cross.
Interesting facts and notes
The choice of title for this chapter is lovely. In typical wizarding fashion, both meanings of "two-faced" are meant in referring to Quirrell: he has been two-faced in that he has been deceiving people and leading a double life, and he has been leading a double life literally, complete with two faces, since his failure to steal the Stone from Gringotts.
"Yes, Severus does seem the type, doesn't he? So useful to have him swooping around like an overgrown bat. Next to him, who would suspect p-p-poor, st-stuttering P-Professor Quirrell?"
And the moral of that story is - taking someone else at face value can be a BIG mistake. This is the only point in the chapter where Quirrell's stutter asserts itself, to make the point that he's been faking it all along to mislead people into underestimating him. The bat reference is one of many in the books which convinced many fans that Snape was a vampire...until Rowling dispelled that idea during an interview.
"She broke my eye contact with you."
Apparently even an experienced wizard like Quirrell - even Quirrell possessed by Voldemort - needs to maintain eye contact to maintain that sort of magical attack.
"Another few seconds and I'd have got you off that broom."
The unnamed spell that Quirrell was using may have been a Hurling Hex; here we know the effect, but not the name or incantation of the spell, whereas for the Hurling Hex we have the name but little other information (PA12).
muttering a counter-curse, trying to save you...
Did Snape know who was casting the curse spell he was fighting? If he knew it was Quirrell, it is hard to believe that he didn't do something about it after the match, either taking care of Quirrell himself or bringing Dumbledore into things. If he didn't know it was Quirrell, certainly Snape, as a staff member of the school, would have sounded the alarm that some unknown person was trying to murder a student. Instead Snape apparently decides to turn this into a one-on-one contest between himself and Quirrell with Harry's life the prize.
Why do you think he wanted to referee your next match?
Did Snape assume that he would be able to defend Harry more easily while trying to referee a complicated game taking place in three dimensions a hundred feet up in the air rather than sitting in the stands, able to watch not only the game but everyone around him? Maybe he wasn't really paying much attention to the game--which would explain the penalty he assigned for no reason whatsoever--and was using the height to be able to watch the crowd for the unknown assailant.
Quirrell snapped his fingers. Ropes sprang out of the air and wrapped themselves tightly around Harry.
Quirrell is not mentioned as having used his wand at all, or as having used any incantation to conjure the ropes.
This mirror is the key to finding the Stone...
This really was a clever hiding place. Unfortunately, Harry managed to get the Stone out of the mirror, something practically no one could have done, and thereby exposed himself to Quirrell's attack and almost gave the Stone away to Voldemort. Dumbledore's plan was undone by Harry's pure heart, the very quality which Harry will eventually use to fight Voldemort. If Harry hadn't turned up, Quirrell would never have been able to get the Stone out of the mirror, after all. By going after the Stone--by playing the hero, in other words--Harry almost ruined the whole plan and gave the victory to Voldemort.
"Tried to frighten me - as though he could, when I had Lord Voldemort on my side..."
Quirrell is the only one of Voldemort's followers as of OP to refer to Voldemort by his grandiose name, instead of the more usual "My Lord" and "the Dark Lord". Of course, since Quirrell is playing host to Voldemort at this point, he's a special case.
I'm presenting it to my master...
So Quirrell wanted to find the Stone, but not use it. He wanted to give it ot Voldemort. According to Dumbledore's explanation later in the Hospital Wing, that "only one who wanted to find the Stone--find it but not use it" could get the Stone out of the mirror, Quirrell should have gotten the Stone at that point. Maybe since he and Voldemort were essentially the same person at that moment, he actually did want to use it.
I met him when I travelled around the world
The timeline of this part of Quirrell's life is a bit tricky to work out. When exactly did this around the world trip take place? How long did Quirrell serve Voldemort? How did he transport Voldemort from Albania to Britain if Voldemort wasn't inhabiting him? From what Hagrid said when he and Harry met Quirrell in the Leaky Cauldron, we can determine that Quirrell had been the Defense Against the Dark Arts teacher before his trip, then again after that trip when his personality had changed dramatically. That means that Quirrell taught DADA for at least two years before the current school year, and for at least one year--the one before the current year--he taught DADA while under the direct influence of the Dark Lord.
"When I failed to steal the stone from Gringotts, he was most displeased. He punished me...decided he would have to keep a closer watch on me..."
That is, Voldemort took possession of Quirrell's body only after the failed attempt to take the Stone from Gringotts. Voldemort was actually being unreasonable to punish Quirrell for that, because the theft failed only because the Stone had been removed by Hagrid by the time Quirrell broke into the bank.
"Use the boy...Use the boy..."
Note that Voldemort says this just after Harry starts trying to think of ways to get himself in front of the mirror; it's quite possible that Voldemort was using Legilimency on Harry at this point.
This is the first of several occasions on which Voldemort approaches a problem by trying to use Harry to solve it. Three years from now, he will arrange to use Harry's blood to give him back the use of a body of his own, and the following year he will manipulate Harry into retrieving the prophecy orb from the Department of Mysteries.
It put its hand into its pocket and pulled out a blood-red stone.
This tallies with the legendary description of a Philosopher's Stone that can turn substances into gold. (Lesser varieties were said to exist that could turn substances into other precious metals, such as silver.)
"He lies...He lies..."
It seems very likely that Voldemort is using Legilimency at this point, because Harry hasn't had much chance to betray himself through body language, even if Voldemort's eyes weren't covered by the turban.
"Now...why don't you give me that Stone in your pocket?"
Legilimency is almost certainly indicated here. Harry only had a chance to walk a few paces after acquiring the Stone before finding himself unable to move; without seeing into Harry's mind, it's hard to say how Voldemort could've guessed that the Stone had wound up in Harry's pocket.
"...not being truly alive, he cannot be killed."
This is disturbing, considering that "Neither can live while the other survives," isn't it?
Even Neville scraped through, his good Herbology mark making up for his abysmal Potions one.
This is one of the earliest mentions of Neville's blossoming talent for Herbology.
Hagrid was there to take them down to the fleet of boats that sailed across the lake
So the students travel back to the station by boat, apparently. In the third book, Harry discovers that the usual method for traveling up to the castle is by carriage, and in the fifth book he learns that these carriages are pulled by invisible horse-like creatures called Thestrals. Starting in the third book, the students ride in these carriages back to the station.
Exceptional character moments
Dumbledore, who has the perfect moment to tell Harry all about Trelawney's first prediction - when Harry actually asked the key question - but let the moment pass because he cared more about allowing Harry to have a childhood than about the fight against Voldemort, at that moment.
Dumbledore's outlook on death, as being but the next great adventure.
Dumbledore, awarding the decisive ten points not to Harry, but to Neville, who had never won any points for Gryffindor before.
Hagrid, presenting Harry with an album of wizarding photographs of Lily and James, since he knew Harry didn't have any pictures of them.
Harry, who could face the Mirror of Erised and want to find the Philosopher's Stone, the key to immortality and wealth, without wanting to use it.
Quirrell, who drops his stuttering, bumbling pose and reveals that he, not Snape, has been after the Stone all along.
Snape, who has actually been trying to protect Harry throughout the school year because he feels an obligation to James Potter, and can't bear to be in debt to a man he hated so much.
"There is no good and evil, there is only power, and those too weak to seek it..." [Quirrell]
"After all, to the well-organized mind, death is but the next great adventure. You know, the Stone was really not such a wonderful thing. As much money and life as you could want! The two things most human beings would choose above all - the trouble is, humans do have a knack of choosing precisely those things which are worst for them." [Dumbledore]
"Always use the proper name for things. Fear of a name increases fear of the thing itself." [Dumbledore]
"The truth." Dumbledore sighed. "It is a beautiful and terrible thing, and should therefore be treated with great caution."
"I always said he was off his rocker," said Ron, looking quite impressed at how mad his hero was.
"Second -- to Miss Hermione Granger...for the use of cool logic in the face of fire, I award Gryffindor House fifty points." [Dumbledore]
"There are all kinds of courage," said Dumbledore, smiling. "It takes a great deal of bravery to stand up to our enemies, but just as much to stand up to our friends. I therefore award ten points to Mr. Neville Longbottom."
It seemed as though life would be back to normal next year, or as normal as it ever was at Hogwarts.
It was Uncle Vernon, still purple-faced, still moustached, still looking furious at the nerve of Harry, carrying an owl in a cage in a station full of ordinary people.