One thing that made me curious while watching a football game on TV was the numbers that appear on the ground. For instance, there's a "1st & 10" beside one team, but there was a "3rd & 6" a few minutes prior (and I think it was beside the other team). What do these numbers mean?
"1st" is the down. In crude terms, a "down" is when a play occurs. A play takes place when a team attempts to advance the ball from its current spot.
"10" is the number of yards needed to obtain a new series of downs (thus, "restarting" at 1st down).
In a possible scenario based on your question: On 3rd (down) and 6 (yards), Team A may have gained 7 yards (re: the first point under Rule 7, Section 3, Article 1 - see below). Thus, Team A is beyond the number of yards needed to obtain a new series of downs. As a result, Team A has 1st (down) and 10 (yards).
As a starting point, Rule 3, Section 9, Article 5 of the 2016 NFL Rulebook defines a first down:
The initial down in each series is the First Down. If it is a charged down, subsequent charged downs are numbered consecutively (i.e., second down, third down, or fourth down) until a new series is declared for either team.
How is a new series declared? Rule 7, Section 3, Articles 1 and 2 describe how for both teams.
ARTICLE 1. NEW SERIES FOR TEAM A
A new series of four scrimmage downs is awarded to Team A when the following conditions exist:
During a given series, the ball is declared dead in possession of Team A while at or beyond the line to gain
Enforcement of a foul by Team B results in the placement of the ball at or beyond the line to gain, or if the enforcement of a foul awards an automatic first down to Team A
If there is a change of possession during the down, and Team A has possession at the end of the down
The kicking team recovers a scrimmage kick anywhere in the field of play after it first has been touched beyond the line of scrimmage by the receivers.
The kicking team legally recovers a free kick.
ARTICLE 2. NEW SERIES FOR TEAM B
A new series of four scrimmage downs is awarded to Team B when the following conditions exist:
Team A fails to advance the ball to the line to gain during a series of downs; or
There is a change of possession during the down, and Team B has possession at the end of the down, or has been awarded possession at the end of the down; or
When the ball is declared dead in possession of the receiving team after a free kick, scrimmage kick, or fair-catch kick.