Each team in the softball league plays each of the other teams exactly once. if 21 games are played, how many teams are in the league?

Each team in the softball league plays each of the other teams exactly once. For every game, there is 2 team playing. The order is not important because  A vs B is same as B vs A
So you just need to makes a combination of 2 that have a result of 21. If there is t number of teams, the number of matches would be:
tC2 = t!/2!(t-2)! = 21
t! / (t-2)! = 21 *2
(t)(t-1)= 42
t^2 -t -42=0
(t-7)(t+6)
t=7 ; t=-6

Excluding the minus result, you got 7 teams.