Rushing has two different meanings in gridiron football

Gridiron football

Gridiron football , sometimes known as North American football, is an umbrella term for related codes of football primarily played in the United States and Canada. The predominant forms of gridiron football are American football and Canadian football...

 (American

American football

American football is a sport played between two teams of eleven with the objective of scoring points by advancing the ball into the opposing team's end zone. Known in the United States simply as football, it may also be referred to informally as gridiron football. The ball can be advanced by...

 and Canadian football

Canadian football

Canadian football is a form of gridiron football played exclusively in Canada in which two teams of 12 players each compete for territorial control of a field of play long and wide attempting to advance a pointed prolate spheroid ball into the opposing team's scoring area...

).

Offense

The first is an action taken by the offensive team that means to advance the ball by running, as opposed to passing

Forward pass

In several forms of football a forward pass is when the ball is thrown in the direction that the offensive team is trying to move, towards the defensive team's goal line...

. A run is technically any play that does not involve a forward pass. It is usually done by the running back

Running back

A running back is a gridiron football position, who is typically lined up in the offensive backfield. The primary roles of a running back are to receive handoffs from the quarterback for a rushing play, to catch passes from out of the backfield, and to block.There are usually one or two running...

, after a hand-off from the quarterback

Quarterback

Quarterback is a position in American and Canadian football. Quarterbacks are members of the offensive team and line up directly behind the offensive line...

, although quarterbacks and receivers can also run. The quarterback will usually run when a passing play has broken down, because there is no receiver open and the field has opened up. A team with an athletic quarterback may regularly call designed run plays for the quarterback, although, because of the risk of injury, this is rare. A wide receiver

Wide receiver

A wide receiver is an offensive position in American and Canadian football, and is the key player in most of the passing plays. Only players in the backfield or the ends on the line are eligible to catch a forward pass. The two players who begin play at the ends of the offensive line are eligible...

 can run on an end-around, reverse, or on a screen pass

Screen pass

A screen pass is a type of play in American football. During a screen pass, a number of things happen concurrently in order to fool the defense into thinking a long pass is being thrown, when in fact the pass is merely a short one, just beyond the defensive linemen. Screens are usually deployed...

. On the screen pass the quarterback will usually throw overarm, and whether it is a run or pass will depend on whether the pass goes forward or backwards. A wide receiver screen is usually intended to be a pass, and if the wideout drops it, it is an incompletion rather than a fumble.

A rushing attempt may also be referred to as a "carry" or "rush", as in "the running back had 20 carries for 100 rushing yards".

Defense

The other, usually called a pass rush, is an action taken by the defensive team that means to charge towards the quarterback or kicker across the line of scrimmage

Line of scrimmage

In American and Canadian football a line of scrimmage is an imaginary transverse line beyond which a team cannot cross until the next play has begun...

. The purpose is tackling

Tackle (football move)

Most forms of football have a move known as a tackle. The primary purposes of tackling is to disposses an opponent of the ball, to stop the player from gaining ground towards goal or to stop them from carrying out what they intend....

, hurrying or flushing the quarterback, or blocking or disrupting a kick. In both college and professional football, getting a strong pass rush is an important skill, as an average quarterback will often be able to be productive, even against a strong secondary, if he has lots of time to pass the ball. Teams will improve their pass rush by blitzing

Blitz (American football)

In American football or Canadian football, a blitz or red dog is when players on or behind the line of scrimmage during a play, are sent across the scrimmage line to the offensive side to try to tackle the quarterback or disrupt his pass attempt...

, which sends additional men at the quarterback. The rules on tackling a quarterback are fairly protective, and very protective in regards of the kicker. This has resulted in most teams sending few men at the kicker, as the risk of drawing a penalty is too great to justify the low chance of blocking a kick.
If the running back takes into the end zone with the defence on him, it is by rule a touchdown.