Rugby looks chaotic if you do not know what you are looking at. Once you have the basics, it is one of the more elegant team sports. The trick is that there are actually two different sports both called rugby, and they have different rules.
The two codes
Rugby split into two versions in 1895 over a dispute about whether players should be paid. The amateur side became rugby union (15 players per side), and the professional breakaway became rugby league (13 players per side). They use the same ball and look superficially similar, but the rules differ in important ways.
In union, possession is contested at every tackle — players from both teams fight for the ball on the ground in a "ruck." Scrums and lineouts also feature heavily.
In league, a tackled player keeps possession but the attacking team is limited to six tackles before they must give the ball back (the "six tackle rule"). Faster, more open, less contested.
How scoring works
Try — grounding the ball over the opposition's goal line. Five points in union, four in league.
Conversion — a kick at goal taken after a try, worth two points.
Penalty kick — three points in union (two in league), taken after a penalty.
Drop goal — a drop-kick during play. Three points in union, one in league.
The rules that matter for following along
You cannot pass the ball forward. All passes go backward or sideways. You can run the ball forward, kick it forward, or be tackled with it.
You must release the ball after being tackled. The tackled player has to let go. The attacking team usually wins it back at the resulting ruck if they have support.
Offside. Players from the kicking team in front of the kicker cannot interfere with play until they are put onside.
Scrums and lineouts. Used to restart play after certain infractions. They look ritualized because they are.
Watching a match
A union match is 80 minutes split into two halves. A league match is also 80 minutes. Start with a Six Nations match (union, Europe, February-March), a Rugby Championship match (union, southern hemisphere, August-October), or an NRL game (league, Australia). The pace of league is easier on a newcomer; the strategy of union is more rewarding once you understand the rules. There is a page with a longer guide on differentiating the two codes.