Typically, you just hear those players on the kicking team referred to as the special teams unit, but there is an unofficial name that is used for those who run down the field to tackle the returner.
In American football, a gunner, also known as a shooter, flyer,
headhunter, or kamikaze, is a player on kickoffs and punts who
specializes in running down the sideline very quickly in an attempt to
tackle the kick or punt returner. Gunners must have several
techniques in order to break away or "shed" blockers, and have good
agility in order to change their running direction quickly. Gunners on
the punt team also must be able to block or catch.
Gunners typically also play positions as defensive backs, wide
receivers, or running back when not on special teams, often as
There's also a specialized role for the player who is specifically tasked with blowing up the blocking on the return.
Tom Powers of Minnesota's Pioneer Press recently highlighted a football position he says is a "you can't pay me enough to do that" job. It's the "wedge buster" - the player who runs down the field after a kickoff and attempts to break through the human wedge a return team builds to block the player returning the ball. "It is the worst job on the football field," Mr. Powers writes. "And a guy has to be exceptionally screwy to be able to do it properly." Heath Farwell, a wedge buster for the Minnesota Vikings, explains: "There's 300-pound guys back there and you're taking on three or four of them, trying to break everybody up. I've seen guys get concussions, get knocked out." Though many, like fellow wedge buster Vinny Ciurciu, have a taste for the work. "Some guys love to blow up the wedge." Hey, it beats working in Scotland.
As far as the return team, there are a few terms used to refer to the blockers as well.
During a kickoff, every player on the return team is called an "upman" with the exception of the one or two designated kickoff returners, who stand furthest away from the starting point of the kicking team.
For punt returns, another term is used.
In American football, jammers try to slow down gunners during punts by preventing them from getting a free release, giving punt returners more time to return punts. A jammer is typically played by a back-up for another position. This position is usually referred to simply as a special teamer, or even a gunner.