 I would also suggest reading articles by authors here at GammonVillage, which include Walter Trice and Doug Zare. These articles are excellent and can span the needs of new players all the way through top players.
Jay Bidal has a series of annotated matches geared towards beginners, which is advised reading and I (Steve Sax) have over 40 articles on archive at GammonVillage which I encourage all students of the game to read. They are the from the "Advanced Angles" series.
Getting back to the basics of the game, if you are successful in getting all of your checkers into you home section of six you may then "bearing-off" (removing checkers from the board).
You must have all of your remaining checkers in your home board to do this. When you roll your dice you may then remove pieces from your home board in correspondence with the number on each die or dice.
If for example you roll a six and four, you may take a piece off the six point and one off the four-point. (enter pos 6a here) If you have no checkers on your six point you may take a piece off the highest point that you have checkers on as a substitute for that highest number. (enter pos 6b here)
This is true for bearing off pieces on lowered numbered points as well. If you roll any number that is greater than the number of point of the highest number you have checkers on you can use that number to take a checker off with. (enter pos 6c here)
You may play either the lower number first or the higher number but if you play both numbers you must even if it is bad for you. (enter pos 7a here and 7A1). You may move the smaller number first even if by doing so you will not move the full number of spaces for the second die. In example 7b the player moves a two first to his four point and then takes a checker off the four point for his movement of the six. Even though he could move all eight spaces he is allowed to make this creative checker play legally and avoid leaving a blot. (enter pos 7b here).
If you can either play the larger number or smaller number but not both numbers then you must play the larger of the two numbers. In this position you would like to hit the blot on the 18-point but it is not legal. You must move (enter pos 7C here) to the 17 point and be subject to being sent back on the bar.
Bearing off against no interference is a straightforward thing but when your opponent has single pieces or points in your home board your bear off can be problematic since many of your moves may be uncomfortable and expose you to leaving blots or becoming awkward. (enter pos 8-1 and 8-2 here).
Another aspect of the game, which I have not mentioned yet, is the doubling cube. As in poker you can play backgammon as a cash game or a tournament game. In either variety there is a way of "raising" which is called the doubling cube.
Anytime it is your turn but before you roll you may turn the doubling cube from the level it is at to double that size. The advantage is that you are playing for twice the stakes but the disadvantage or that is that your opponent then owns the cube and is the only one who can then double the stakes "raise".
The cube begins the game in the middle and stays there until someone turns the cube before their turn. For the remainder of the game the cube will be in possession of one player or another.
To a degree this is like pot limit as you can raise the pot but there are differences in that you can win or lose more than the level of the cube because of bonuses called gammons and backgammons.
The gammon bonus occurs when you take all of your checkers off and your opponent has taken no checkers off. That bonus is that whatever the level of the cube is on you win double that number of points.
Another bonus even more rewarding is the backgammon bonus, which is a triple game. That occurs when you take all your checkers off and your opponent has no checkers off and at least one checker in your home quadrant or on the bar.
In tournament play you do not need to turn the cube to achieve those bonuses but in money play you must turn the cube or have taken the double to achieve a gammon or backgammon bonus. That rule is called the "Jacoby rule" which was invented by Oswald Jacoby one of the all time great games players.
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