"Three Dementor attacks in a week, and all Romilda Vane does is ask me if it's true you've got a Hippogriff tattooed across your chest." -- Ginny Weasley
Hermione finds out about Eileen Prince in the library and she thinks Eileen may be connected to the Half-Blood Prince. Dumbledore asks Harry to come to his office immediately. On the way there, Harry runs into Trelawney, learning from her both that Malfoy is incredibly happy and that Snape was the one who overheard the prophecy. Dumbledore asks Harry to accompany him on a quest for the Horcrux, but first Harry confronts him again about Snape; finally, Harry runs off to fetch his invibility cloak, giving Ron and Hermione his map and Felix Felicis, and he and Dumbledore apparate to the seaside.
Calendar and Dates
All we know about this chapter is that, aside from a brief conversation at the beginning, the entire thing takes place in one afternoon in June just prior to O.W.L.s.
Interesting facts and notes
the lightning-struck tower
A real Tarot card, labelled as "The Tower" in a Waite-Ryder Tarot deck but also known by the name Trelawney uses here. Her interpretation is pretty standard, as far as I know.
the centaur - knows nothing of cartomancy
That is, the art of divining the future through the use of cards.
I must confess that, at first, I thought he seemed ill-disposed toward Divination
In fact, by Dumbledore's own admission at the end of Harry's fifth year, he had at the time been considering discontinuing Divination as a subject taught at Hogwarts (OP37).
he himself was seeking a job at that time
Snape said during Umbridge's evaluation of his class (OP17) that as of that time, he had taught at Hogwarts for fourteen years, which would place the start of his teaching career sometime after Harry's birth since Umbridge's interrogation took place when Harry was 15.
he ran past her, around the corner into Dumbledore's corridor
That is, Harry is still on the seventh floor, since he and Trelawney were just outside the Room of Requirement. This conflicts with statements about the location of Dumbledore's office in previous books.
Exceptional character moments
Trelawney (unknowingly) speaks to Harry about the night she foretold the prophecy about "the Chosen One". Harry finds out that Snape told Voldemort about the prophecy.
Trelawney correctly predicts the events that will happen later that night using her tarot cards; She reads them off: Lightning-struck tower, calamity, disaster.
"And what did you tell her Ron's got?"
"A Pygmy Puff, but I didn't say where."
"You filthy hypocrite! What about you and Lavender, thrashing around like a pair of eels all over the place?"
"'Thanks', said Ron. 'Er - why do I need socks?'"