A game of football has cycles of downs where a team has to travel a distance of 10 yards towards the end zone. A third down is the team's third chance out of four at maximum to travel those 10 yards. If a team gets to a third down it means that it has failed to travel the 10 yards in the first and second downs combined. A third down situation puts the attacking team under some pressure; that is because most of the time if a team fails to meet the remaining distance in the third down, the ball is returned to the opponent, or a field goal kick is attempted, as fourth down plays or rushes are rather risky.
You may hear expressions such as "Third and 6". That means that the attacking team is at its third attempt of traveling 10 yards, its third down; and still has to travel six yards, as it traveled a total of four yards in the first and second downs combined. You may also hear "Third and 15". A team can also travel backward in the field, and in that case, the attacking team traveled five yards back in the two previous downs combined, and so it has to travel fifteen yards on its third down.
3rd down is the penultimate down of the down cycle before 4th down. For most teams in a standard game, 3rd down is the final down you get to run a play because most teams punt the football to the other team on 4th down. In 3rd and long scenarios (situations where the 1st down line is ten or more yards away), passing is imperative unless you're trying to fool the defense by running the ball. In 3rd and short situations (situations where the 1st down line is less than three yards away), running and passing are both viable options for the offense.
If the offense does not reach the 1st down marker on 3rd down, it is now 4th down. Depending on the situation, the offense may elect to go for the 1st down on 4th down, which is very risky.