Speed is used to determine which fighter will be first to hit (either you or your opponent). Depending on whos speed is higher, they will be at an advantage as they will be first to strike. Speed-heavy fighters are good as if you manage to land a hit first and defeat them, you take 1 less hit worth of damage.
Speed can be tricky when it comes to bosses though..
Bosses will always attack either before, or after, each of your fighters. If the speed of your fighter(s) is/are faster, the attacks will resolve as follows: fighter1 (bottom fighter on your team), boss attacks fighter1, fighter2 (middle fighter on your team), boss attacks fighter2, fighter3 (top fighter on your team), boss attacks fighter3, and will then repeat until the boss or your team is defeated.
Should one of your fighters be slower than the boss, the boss will appear to attack twice in a row. Lets say, that fighter1 and fighter3 on your team are faster than the boss, but fighter2 is not. The 'round' would resolve as follows; fighter1, boss attacks fighter1, boss attacks fighter2, fighter2, fighter3, boss attacks fighter3, repeat.