The Quidditch position responsible for catching the Golden Snitch is the Seeker. Each team has one Seeker, whose specialized job is to locate and capture the tiny, fast-moving golden ball.
Catching the Snitch normally ends the match and awards the successful Seeker's team 150 points. Because that bonus is so large, Seekers can decide a game even after their team has played poorly. However, a Snitch catch does not automatically guarantee victory: in the 1994 Quidditch World Cup final, Viktor Krum caught the Snitch for Bulgaria, but Ireland was already far enough ahead to win by 10 points.
Harry Potter plays Seeker for the Gryffindor team at Hogwarts, making the position especially prominent in the books and films. Other notable Seekers include Draco Malfoy, Cedric Diggory, Cho Chang, and Viktor Krum.
The other positions have different duties. Chasers handle the Quaffle and score goals, the Keeper defends the hoops, and Beaters use bats to protect teammates from Bludgers. None of those positions catches the Snitch.