I know there was already some post about Trelawney, but I just wanted to sum it up.
Her two known prediction that came true both connected to Voldemort:
“It will happen tonight. The Dark Lord lies alone and friendless, abandoned by his followers. His servant has been chained these twelve years. Tonight, before midnight… the servant will break free and set out to rejoin his master. The Dark Lord will rise again with his servant’s aid, greater and more terrible than ever he was. Tonight… before midnight…the servant…will set out…to rejoin…his master…." (POA )
Pettigrew escapes and rejoins Voldemort and a year later he helps him get his body back
"The one with the power to vanquish the Dark Lord approaches.…born to those who have thrice defied him, born as the seventh month dies…and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not.…and either must die at the hand of the other for neither can live while the other survives.…the one with the power to vanquish the Dark Lord will be born as the seventh month dies…" (OOTP)
I don’t think it has to be explained.
Her minor predictions:
The big black dog she sees in Harry’s cup either is not the Zordo but represents Sirius, ( the dog star, Sirius’s animagus form ) or is the Zordo, the omen of death looming over Harry till his actual death in 1998.
“Professor Trelawney was now teaching them palmistry, and she lost no time in informing Harry that he had the shortest life line she had ever seen.” GOF
Also Harry dies at 17.
"He therefore had to endure over an hour of Professor Trelawney, who spent half the lesson telling everyone that the position of Mars with relation to Saturn at that moment meant that people born in July were in great danger of sudden, violent deaths. "Well, that’s good," said Harry loudly, his temper getting the better of him, "just as long as it’s not drawn-out. I don’t want to suffer." GOF
Harry’s death scene quite violent but sudden and painless.
13 people at the table:
Later Trelawney again met Harry in the corridors, and this time complains that Dumbledore had been repeatedly dismissing her warnings of impending disaster. She looked at some Tarot cards and saw,
"… the lightning-struck tower… Calamity. Disaster. Coming nearer all the time…".
This seems to predict the events of Dumbledore’s death on the Astronomy Tower. The Lightning-Struck Tower tarot card (The Tower) is generally considered representative of disaster or life-altering change.
"Yes," said Professor Trelawney, nodding impressively, "it comes, ever closer, it circles overhead like a vulture, ever lower… ever lower over the castle… .” GOF
From Trelawney to Harry:
"You are preoccupied, my dear," she said mournfully to Harry. "My inner eye sees past your brave face to the troubled soul within. And I regret to say that your worries are not baseless. I see difficult times ahead for you, alas… most difficult… I fear the thing you dread will indeed come to pass… and perhaps sooner than you think…” GOF
Before the class Harry tells Hermione that he is worried about Trelawney’s prophecy about Pettigrew and that Voldemort will return.
“I was saying that Saturn was surely in a position of power in the heavens at the moment of your birth…your dark hair…your mean stature…tragic losses so young in life…I think I am right in saying, my dear, that you were born in midwinter?”
Later on we learn that Voldemort was born in mid winter, and that part of his soul - what Trelawney felt - was in Harry.
My dear, you were undoubtedly stimulated by the extraordinary clairvoyant vibrations of my room!” said Professor Trelawney. “If you leave now, you may lose the opportunity to see further than you have ever -“ GOF
Right after Harry falls aslep and gets a vision about Voldemort.
Harry and Ron’s predictions:
From Harry to Ron ”so you’re going to suffer but be very happy….” POA,
Ron is attackted by Sirius, loses his rat but at the end of the book he is happy because Harry gets back his godfather and it turns out his rat was Pettigrew the traitor.
From Ron to Harry: "There’s a blob a bit like a bowler hat," he said. "Maybe you’re going to work for the Ministry of Magic…” POA
Harry becames an Auror working for the Ministry.
“‘A windfall, unexpected gold.” POA
Harry wins the Twizadry Tournament and the prize is gold, he inherits Sirius’s gold too.
At the final Divination exam Harry has to look into the crystal ball, he does not see anything but he lies that he can see a hoppogriff flying away free and as we know he frees Buckbeak a few hours later.
Harry and Ron making up their homework:
“I will be in danger of- er - burns.” / "Lose a treasured possession," / "Why don’t you get stabbed in the back by someone you thought was a friend?” / I think I’ll come off worst in a fight.” / “Okay, I’ll lose a bet.”
During the Triwizard Tournament all the predictions come true: Harry gets burned by the Hungarian Horntail / looses a treasured possesion meaning his friends getting kidnapped by the mermaids / Ron thinks he is a liar and ignores him / looses in a duel against Voldemort / Bagman loses a bet on Harry
sources: Harry Potter Wiki , Harry Potter books