Rugby Football (usually just called "rugby") is named after Rugby School, Warwickshire, where the modern game was developed. However the modern game descends from much older variations of football games. And, descending from modern rugby are Australian Rules Rugby, American Football, and Canadian Football. Rugby Union and Rugby League are the two main variations of the game, and both include professional and amateur teams.
In rugby, ground is gained only by running or kicking the ball; the forward pass is not allowed (as it is in American Football.) The Union version has teams of 15 players; League, 13. (American Football has 11 players.) The following additional rules will refer only to the Union Version.
As with American Football, the ball may be advanced by either carrying it or kicking it. Usually, in rugby, the ball is carried rather than kicked. In American Football, the ball is seldom kicked except in specific situations, leading to the erroneous belief that the rules don't even allow for kicking on any play. In American Football, the ball is either carried or "passed" forward. In rugby a pass may not be forward (advancing toward the goal. In Rugby, a pass means in a non-forward direction. Such a "pass" (sideways or backwards) is called a "lateral" in American Football. Both games allow any number of non-forward passes during a play.
In rugby, play is continuous and possession of the ball is contested after each tackle, and is thus more continually physical, rather than being physical in "spurts" like American Football. In American Football, the possessing team is simply allowed 4 tries to advance the ball 10 yards, and play stops between tries. Although opposing players certainly try to take the ball away from the runner in American Football, if the opportunity arises, the object is more to tackle him before he gains too many yards. (The possession of the ball changes if 10 yards are not gained in 4 tries.)
In rugby, a tackled player must immediately give up the ball, either by passing it or simply releasing it if no pass is possible. At that point the loose ball is contested, just as it would be in American Football if a tackled player were to accidentally lose control of the ball on the way down. An American might go so far as to say that, in rugby, the ball is "fumbled" each and every tackle, and a fight for possession ensues each time.
In rugby, the ball-runner's team mates must stay behind him. Thus American Football-style "blocking" is not allowed in rugby. (In American Football, players may run in front of the ball carrier and "block" defensive players for him.)
Points are scored in rugby (or American Football) by either advancing the ball over the goal line, or by kicking the ball over the goal crossbar. In American Football, and additional method of scoring is allowed by the legal reception of a forward pass while standing in the goal area.
Infraction of rules in American Football are enforced by taking away yardage from the team in possession, or giving yardage to them if the foul was on the defensive team. Different amounts of yardage penalties are assigned for various penalties; 5 yards for minor infractions, 10 yards (rare) for certain specific infractions, and 15 yards (common) for major infractions.
The field: A rugby field (pitch) dimensions are (apparently, according to the rules I read) flexible. "No more than 70 meters in width and no more than 100 meters in length." May it be smaller? Guess so. An American Football field is exactly 100 yards long from goal line to goal line (with an additional 10 yards on each end which serve as "end zones"; and 160 feet wide. A goal post is centered at the very back of each end zone, in American Football. In rugby, the goal posts are on the goal lines (try lines) themselves. I think. And there is an area at each end of the field, beyond the try line, which may not be less than 10 meters deep. Apparently it may be more.
The ball: In rugby the ball is a "prolate spheroid". An American Football is almost a prolate spheroid. (A rugby ball is fatter.) In both cases, the balls are shaped the way they are because that is how pig bladders are shaped, though pig bladders are no longer used.
Conclusion: Rugby is a lot more complicated than American Football.