British colonists and the British military in Canada brought rugby to North America. It along with association football became popular in American and Canadian universities. At the time soccer and rugby were not as differentiated as they are now and teams would negotiate the rules before playing a game. The sport of American football evolved from these intercollegiate games.
Meanwhile in England a schism developed between those who favoured strict amateurism and those who felt that players should be compensated for time taken off work to play rugby. In 1895 this resulted in the formation of a break-away sport, rugby league, the rules of the two codes of rugby union and league would themselves diverge.
A try is the rugby equivalent of a touchdown. Unlike American football, both codes of rugby require the ball to be grounded, in American football it is sufficient for the player carrying the ball to enter the endzone (in-goal area). In American football a touchdown scores 6 points, in rugby union it is worth 5 points and in rugby league it is worth 4 points.
In all three games, following a try / touchdown, there is the opportunity to score additional points by kicking the ball between the posts and over the bar. In American football this is called an extra point (worth 1 point), in both codes of rugby it is known as a conversion (worth 2 points). There are two key differences between an extra point and a conversion, conversions cannot be charged down like an extra point attempt but they must be taken from the same position as the try was scored. Hence it is important to score under the posts rather than in the corner which makes for a difficult kick.
In American football teams often opt to go for a field goal (worth 3 points) rather than attempt a touchdown. The rugby equivalent is a drop goal (worth 3 points in union and only one in league). The key difference between a field goal and a drop goal is that a field goal attempt is normally kicked with a team-mate holding the ball, whereas in rugby the ball must hit the ground and be kicked from a half-volley.
A similar concept in rugby is the penalty goal. Following the award of the penalty, the attacking team may opt to kick for goal rather than advance the ball by hand or punting. This scores 3 points in union and 2 points in league. The penalty goal is similar to a field goal in American football in that the ball is kicked from the ground and may be held by a team-mate, but it cannot be charged. There is no direct equivalent to a penalty goal in American football.
American football has one further method of scoring which does not exist in rugby. If a ball carrier is tackled in their own endzone (in-goal area) with the ball then this results in a safety which scores 2 points. In rugby union this does not score any points but causes the team in possession to kick the ball back to the opposition from their 22 metre line, the rule is similar in rugby league but the kick takes place from under the posts.
In American football, one team retains possession, unless there is a fumble or interception, until they have used up all four downs. Rugby league is very similar with the team in possession having six tackles before having to hand over possession. The key difference is that in American football advancing the ball ten yards earns a new set of downs whereas there is no automatic way of earning a new set of tackles in rugby league.
Rugby union is quite different being based on the 'right to contest possession'. A team in possession does not need to surrender possession whilst they are able to keep the ball. Whilst American footballers and rugby leaguers aren't allowed to try to disposses the ball carrier between tackles, rugby union players are allowed to win possession during open play.
In all three codes, it is normal to advance the ball by carrying it in hand but tactical kicking is an important aspect of play. In American football or rugby league, it is normal to punt on the last down / tackle but in rugby union with no limited tackle system a kick can take place at any phase of play.
In rugby, forward passes are not allowed and the only form of passing is what would be called a lateral in American football. Only one pass per down may be thrown forwards in American football but any number of laterals may be thrown, in practice laterals are fairly unusual in American football.