What is a forward pass? A forward pass happens whenever the offense throws the football towards their opponent's endzone. The player throwing the ball must be behind the line of scrimmage and he must be throwing towards an "eligible receiver" (basically, a fancy way of saying any player who is not an offensive lineman; eligible receivers also include all players on defense). If the ball is tapped or touched by an eligible receiver, then the ball is live and can be touched by any player.
What's the difference between forward passes and backwards passes? There can only be one forward pass per play, whereas there can be unlimited numbers of backwards or lateral passes. Sometimes, you will see an "illegal forward pass" called when the offense is trying to throw the ball laterally across the field, but accidentally throws it forward for a second time, like in the play below. If this happens, there is a 5-yard penalty. The down is reset, and replayed again.