How does the downs system work in American football?
A team has four attempts to advance the ball ten yards. Succeeding resets the count; failing hands the ball to the opposition. Almost every tactical decision in the sport follows from this.
The mechanics. The offence gets four downs to gain ten yards from where the sequence started. Gain ten at any point and it is a first down — the count resets to first down with ten yards to go, and they continue.
This is why commentary describes situations as "third and four" — third down, four yards still needed.
Why fourth down is the interesting one. Failing to make the distance on fourth down means the opposition takes possession at that spot on the field, which is usually terrible field position. So teams almost always use fourth down differently:
Punt — kick the ball downfield to force the opposition to start further back. The standard choice when out of field goal range.
Field goal attempt — kick between the posts for three points, viable within range.
Go for it — attempt to make the distance. Historically rare, and now considerably more common, because analytics demonstrated that coaches were punting far too often. The expected value of converting frequently exceeds the value of the field position surrendered, and the shift in behaviour over the past decade is one of the clearest examples of statistics changing a sport's tactics.
Scoring, for context: a touchdown is six points, followed by a conversion attempt worth one (a kick) or two (a play from close range). A field goal is three. A safety, tackling the opposition in their own end zone, is two to the defending team.
Why the clock matters so much. The clock stops for incomplete passes, players going out of bounds, and various other situations. This creates the endgame tactics the sport is known for — deliberately throwing incomplete passes, running out of bounds, and the two-minute drill.
Why a game takes three hours for sixty minutes of play: the clock stops constantly, and there is a break between every play while teams re-form.