On Harry's first trip to Diagon Alley, he kept hearing and seeing mentions of something called Quidditch. Like everything else in the magic world, Harry was completely lost on that. Hagrid and Ron attempted to explain the game to him, but it wasn't until he actually played it that he understood.
Quidditch is a game played on brooms. Each team has seven players: three chasers, two beaters, one goal keeper and one seeker. Chasers can score points by passing the ball through the goal. There is no time limit on the game as it only ends when the snitch is caught by the seeker.
On Harry's first day of flying class at Hogwarts, he and Draco got into a tussle over a remembrall that belonged to Neville Longbottom. Harry chased after Draco on his broom, despite having no prior knowledge of how to fly. He caught the ball, and the attention of Professor McGonngall. Moments later, he was being shown to Oliver Wood, captain of the Gryffindor Quidditch team. Harry became the youngest player and seeker at Hogwarts in over a hundred years.
First Year: Defeated Slytherin, Hufflepuff. Did not play in the final game against Ravenclaw.
Second Year: Defeated Sytherin, but not without injuries. Just before the Hufflepuff game was to start, it was canceled due to attacks on students. No more Quidditch games were played that year.
Third Year: Due to the influence of the dementors, Harry fell hard off his broom, ending Gryffindor's chance at winning the match. His beloved Nimbus 2000 was also a casualty of the fall. With a shiny new Firebolt, Gryffindor defeated Ravenclaw in February. Gryffindor next defeated Slytherin in the Cup final, finally earning them the best award of all.
Fourth Year: No Quidditch games were played during Harry's fourth year due to the Triwizard Tournament. Harry did make good use of his broom in the first challenge of the Tournament.
Fifth Year: Gryffindor won their first game of the season, against Slytherin. Harry was banned from Quidditch by Umbridge, so he did not play the rest of the year.
Sixth Year: Harry was made Quidditch captain of the Gryffindor team! Under Harry's leadership they defeated Slytherin and lost to Hufflepuff. Harry was not able to play the final match (against Ravenclaw) due to detention. However, the team won without him, earning them the Cup.
Harry's first broomstick was the Nimbus 2000. After McGongall invited Harry to the Gryffindor team, he received this through the mail. It served him faithfully until his third year. It shattered into bits when he fell from the sky after an attack by the dementors.
Harry's current broomstick is the Firebolt, the fastest currently on the market. This broomstick was the source of a great mystery when he received it on Christmas day with no card. He later learned it was from his godfather as a present to make up for lost years of present giving. It can accelerate to 150 miles per hour in 10 seconds.