Each team has four downs to move the ball 10 yards. If it is able to do that, the team is awarded a new set of downs. First down is the first of those four attempts.
When a team is able to move the ball upfield by 10 or more yards, it is awarded a new set of downs--that is referred to as "gaining a first down", as the team now starts playing a first down again having four chances to move the ball 10 yards.