The offensive team [the team in possession of the ball] has four attempts, or chances, to move the ball ten yards. These attempts are called downs.  From the time the ball is snapped, or put into play, to the time the play is whistled over by the officials, is considered one down.  Once the ball has moved ten yards or more from the starting point of play, the team gets a fresh set of four new downs to try to move the ball another ten yards. If the ball doesn’t move ten yards forward in four downs, the other team gets possession of the ball (the offense also has the option of kicking [punting] the ball to the other team on the fourth down).

The first play of a series is called first and 10 because it is the first down and 10 yards are needed to receive a new set of four downs.

Suppose on the first play, the team on offense picks up three yards. The next play would then be second and 7, because it is the second play of the set and they still need seven yards to get a first down [and a new set of four downs].

If they were to pick up six yards on the second play it would leave them one yard shy of a first down, therefore setting up a third and 1 situation because it would be the third play of the series and they would still need one yard to get a first down.

If the team with the ball can pick up one yard or more on the third down play, then they will be given a first down, which means they get to start all over with a new set of four downs.

A team can continue moving the football down the field as long as they continue to pick up first downs.

On occasion the yards to go may be a number other than 10, due to a penalty.  

Third down with a significant distance to go (usually over 15 or 20 yards) is known as third and long.  This term is also often used in everyday life as a metaphor for a desperate situation that demands risky actions be taken.  

Fourth down with less than one yard to go is referred to as fourth and inches. This is often used in tense situations where the offense is tempted to scrimmage the ball rather than kick for a chance to get another first down.  

First down, where the goal line is the line to gain, is known as first and goal.  A team cannot make another first down (barring a defensive penalty) without actually scoring.  

Fourth-down Strategies
If a team fails to gain the required yardage on third down, a team basically has 3 options on fourth down:
• The majority of the time, teams will "punt" the ball away on fourth down. A punt is simply a form of kicking the ball that gives possession of the ball to the other team, but also pushes them back considerably farther away from the end zone.
• A team can "go for it" on fourth down and try to pick up the remaining yardage, but they run the risk of turning the ball over to the other team if they do not get to the first down marker. If they do not get the required yardage, the other team takes possession of the ball at the spot of the last tackle and now has four downs to move 10 yards back in the other direction.
• Another option is to kick a field goal. If a team feels they are close enough to kick the ball between the upright bars of the goal post in their opponent's endzone, they may attempt a field goal, which is worth three points.

After a Score
• After a team scores via a touchdown or field goal, they kick off to the other team and the process begins all over again.

Ladies, I hope this info makes sense.  Watching football on Sundays is a great way to spend time with the man you love – and believe me, he will get a kick out of you taking interest in and enjoying football!! 

Source: Wikipedia

Football 101 - Uniform Numbers
Football 101 - Scoring 
Football 101 – Offense & Defense