Madam Hooch instructs the First Years in the basics of flying, using twenty of the school’s sub-par brooms. Neville injures himself falling off his his broom, and Madam Hooch escorts him to the Hospital Wing. While the class is thus unsupervised, Harry and Draco fight it out on brooms over Neville’s Remembrall, which Draco took.
Harry is subsequently made Seeker on the Gryffindor Quidditch team by Professor McGonagall who had seen him flying after the falling Remembrall and realized that he was a natural.
This happens on the second Thursday of the school term.
"...they spotted a notice pinned up in the Gryffindor common room that made them all groan. Flying lessons would be starting on Thursday..."
The date can be determined from the fact that this is the Thursday of the second week of school, it must logically be the 12th, since classes started on the 2nd.
How do we know that this is the Thursday of the second week? Because when Harry is following McGonagall away from his flying lesson, he is afraid that he will be expelled. He thinks to himself:
"Now he'd done it. He hadn't even lasted two weeks."
However, when Fred and George hear about this, they say:
"I tell you, we're going to win that Quidditch Cup for sure this year...We haven't won since Charlie left, but this year's team is going to be brilliant."
According to Rowling's statement that Charlie is three years older than Percy, Charlie just left two months before. There's been no time for them not to have won "since Charlie left." They haven't PLAYED since Charlie left. Again, this is an error in the text unless, as some have speculated, Charlie left school before the end of his seven years, perhaps to go study dragons.
All the Slytherin first-years are supposed to be present, but some (like Crabbe and Goyle) have no lines and are not mentioned by name. These silent Slytherin seldom speak up in Potions, either. They include Millicent Bulstrode, Theodore Nott, Blaise Zabini, and at least one other Slytherin girl. (Since Pansy's gang of Slytherin girls is often mentioned, we know that at least one girl other than Millicent must be in Slytherin in their year.)
Approximately twenty brooms were waiting for students to use. And therefore, there are twenty students between Gryffindor and Slytherin. We are not told if this relatively small number of students is typical of every year at Hogwarts or if this year's Gryffindor and Slytherin classes are remarkably small in numbers. The complete list of students in Harry's year was shown during the "Harry Potter and Me" television special (HPM). Some names are impossible to determine from the image shown on that special.
Neville fell over twenty feet. Granted, he landed on grass, but still. Here we see that uncanny ability of wizards to avoid serious injury in situations that would probably kill a Muggle. Remember that his family learned that he was magical by dropping him out of a second-story window.
This is an interesting scene for Parvati, who usually plays an even smaller role in things. True to her Gryffindor nature, she is quick to stick up for Neville when he is being picked on. Additionally, many readers have pointed out that when Pansy Parkinson teases Parvati for this, she calls her by her first name. Pansy always called Hermione "Granger," so it is interesting that she might be on a first name basis with Parvati.