"So what would you like me to do? Magic myself popular? Conjure myself into a new house? Transfigure myself into a better student?"
-- Albus Potter (CC1.4)
When he begins Hogwarts he is quite nervous about being sorted into Slytherin, thanks in large part to the goading of his older brother James (DH/e). Albus worries that it is not a house of "brave wizards", but Harry says that he was named after a Slytherin who was one of the bravest wizards he knew (CC1.2). Harry further reassures Albus that the Sorting Hat will take his personal choice into account if he wants it to (DH/e).
Albus is Neville Longbottom's godson (Pm).
When he was six years old, his best friend was Rose Granger-Weasley (CC1.10).
Albus attended the 2014 Quidditch World Cup Final with his family, where his father introduced him and his brother to Viktor Krum. He was the only one of the Potter family supporting the Brazil National Team, as his hero was the Brazilian Chaser Gonçalo Flores (Pm). However, by the time he starts at Hogwarts two years later, he and his best friend both despise Quidditch (CC1.4).
He meets Scorpius Malfoy on the Hogwart's Express (CC1.3) and, recognising that they both have issues with their famous parents, they sit together and eventually become best friends. To the shock of the other students and the delight of Scorpius, Albus is sorted into Slytherin (CC1.4). He and Scorpius are kindred spirits - they both carry a burden stemming from their family identity (CC2.6) and feel that they are misfits and "losers" in comparison with the other students at Hogwarts (CC1.4, CC2.8, CC2.16). His father Harry, in particular, could not bring himself to accept that Albus was unpopular and uncooperative in school and he questioned why Albus was content to have only the one friend (CC1.4). The father-son relationship continues to deteriorate (from bad to worse) during Albus’s first few years at Hogwarts (CC1.7).
Prior to leaving for his fourth year, Albus hears about a confiscated Experimental Time Turner while eavesdropping during Amos Diggory’s visit to their home (CC1.6). When he later has confirmation of its existence, Albus decided that he wants to use it to "save" Cedric Diggory - something his father had refused to countenance. He then persuades Scorpius to jump off the Hogwart's Express (CC1.11) and, together with Amos’s niece Delphi, they steal the Time Turner from the office of the Minister for Magic. His plan is to prevent Cedric from being killed as a "spare" at the end of the Triwizard Tournament (CC1.6, GF32).
Their trips back into time to change history were not a success. They first try to prevent Cedric from achieving a high score during the First Task with dragons (CC2.6, CC2.7, GF20). History changed, but not in the ways they expected (CC2.9). Albus and Scorpius made a second attempt by trying to humiliate Cedric during the Second Task underwater (CC2.20, GF26), but this intervention caused history and their own lives to go horribly wrong (CC3).
Left stranded in time by Delphi during the Third Task (CC3.20, GF31), Albus ingeniously sends a message to his father in the future. The grown-ups - Draco Malfoy, Harry Potter, Ginny Weasley, Hermione Granger and Ron Weasley - use the Golden Time-Turner and work together with Albus and Scorpius to ensure that time and history remain unchanged. Following these events, the relationship and understanding between Albus and Harry improves (CC4).
Father: Harry Potter
Mother: Ginny Weasley
Siblings: James Potter, Lily Potter
Supposed to be good at Potions. Enthusiastic and impulsive.
Other canon notes and references
Albus is mercilessly teased by his older brother James and ridiculed by other students. When he fails to control his broom in a flying lesson he is called the "Slytherin Squib" (CC1.4).
Albus is Latin for "white" and, from the same root, Albion is an ancient name for Britain and Albinus is a Roman family name (Wikipedia).
Severus may derive from the word severe, meaning "strict or harsh". The emperor of Rome from 193–211 was Septimus Severus (Wikipedia) and there are also several saints with the name Severus (Catholic Online).
One reason that many fans guessed that Albus will be a Slytherin is that his initials spell out the word asp, a type of snake.
Is Albus the "Cursed Child" of the title? He finds that living up to the expectations of his name - and his father's fame - means he feels that no one except Scorpius Malfoy sees him as his own person. Without his best friend he would be having an extremely unhappy and lonely time at Hogwarts. Albus has no choice about whose son he is, and because he disagrees with Harry's choices, he makes a decision to try and correct an injustice. He becomes a teenager with a crusade and thus more misunderstood than cursed.
From the Web
Screenshots of the Daily Prophet's coverage of the 2014 Quidditch World Cup: http://imgur.com/a/AXutv
About the play: http://www.harrypottertheplay.com
Pottermore news: http://www.pottermore.com/cursed-child