The basic point of rugby is the same as American Football - but in rugby you must physically touch the ball down in the 'endzone'.
Unlike American Football, rugby does not stop once you are tackled. The tackle creates a free-for-all called a 'ruck', or 'breakdown' where anyone on either team can compete for the ball (according to the laws of the game). This is probably the single defining aspect of rugby union that makes it unique amongst other contact sports.
There are 4 other contests for posession besides the ruck. One is obvious - if the ball is bouncing around or in the air then anyone can get it provided they are 'onside'. Another, a 'maul', is similar to a ruck but takes place when a player is tackled and does't make it to ground. This situation comes with its own specific laws. The other 2 contests are set pieces, and take place after the ball has gone dead. A 'scrum' involves the two teams' 8 'forwards' pushing against each other. A 'lineout' happens when the ball goes off the side, and is like a Football throw-in, except players compete for the ball like the restart in basketball. The team that takes the ball into a ruck/ maul, or throws in, or puts the ball into the scrum, is normally expected to get it back.
Another important thing is that the ball can only be passed backwards, which means that the attacking side will generally have everyone lined up behind the ball - unlike American Football or Football where players are all over the pitch. However, you are allowed to kick the ball forwards, provided your teammates who chase the kick are behind you when you kick it. Knocking the ball forwards is the same as a forward pass - illegal. When this happens, possession is handed over to the other team who put the ball into a scrum.
You may only tackle someone in rugby if they are carrying the ball. Blocking is not allowed. Furthermore, you must tackle with your arms wrapped fully around the player, and you must tackle below shoulder height.
Scoring in open play: touching the ball down into the endzone earns you 5 points and is called a 'try'. After this, you get an extra point stlye field goal attempt called a 'conversion', taken in line with where the ball was put down, and worth 2 points. Instead of going for a try, you can also kick a 'drop goal', worth 3 points.
The most common way of scoring, however, is via penalty field goals, called 'penalties'. There are various offences you can commit around the field that result in penalties, and with penalties you have numerous options of what to do. If the location of the penalty is within range of the goalposts, your kicker may decide to take a field goal (in the same way as a conversion). If he kicks it over, his team get 3 points.
Hope that's of use. Try watching some videos on youtube.