This is the best game to play with dominoes, ever. The rules are painfully simple, but allow a ridiculous amount of strategy to be applied. Also, luck is a factor.
That's great, but how do we play?
The game needs at least two players, and is traditionally played with the European style of dominoes (sets that have blanks, and each tile is unique). I've only played with sets that were doublesix, though I've heard that games with doublenines are played to allow more players. It would make sense to assume that even more people could play with doubletwelves, and so on. Most of the rules are debatable, and will vary by location, culture, intoxication, etc. However, the following rules should be hard and fast, or else the game just falls apart.
 Everyone starts with seven tiles.
 If anybody has more than three doubles, everybody redraws tiles and you start over.
 For the first hand, the person with the highest double plays first, playing that double. The highest double is determined by a roll call of sorts. Say you're playing with doubletwelves. Someone says "Twelves?" Nobody plays. "Elevens?" ... and so on, until somebody has that double and plays it.
 For each successive hand, the winner of the previous hand starts play with any tile that they have, double or not.
 The first double that is played (in the first hand, it is the first tile) is the "spinner", and is the only tile that will ever be branched from. We'll go into more about this later.
 You may not branch off of a spinner unless it has a tile on each side already.
 If you cannot play, you must draw tiles from the leftovers (sometimes called the "bone yard") until a playable tile is possessed. If there are no leftover tiles, you knock the table and are skipped.
 The first player to run out of tiles wins the hand.
 The winner is the first person to reach a set amount of points (usually 150 or higher, in increments of 50).
What's that about a spinner?
Best shown with an illustration. In the interests of keeping this writeup in a manageable space, I will use numbers to signify the amount of dots on each section of tile.
Let's say the game has been played this far:
++ ++
+++ 1 ++ 5 ++
 3  6  6  1  1  5  5  4 
+++ 1 ++ 5 ++
++ ++
Now, let's give a bit of history and say that the doublefive was played before the doubleone. In this case, a legal move would be:
++
 2 

 5 
++ ++
+++ 1 ++ 5 ++
 3  6  6  1  1  5  5  4 
+++ 1 ++ 5 ++
++ ++
... and an illegal move would be:
++
 2 

 1 
++ ++
+++ 1 ++ 5 ++
 3  6  6  1  1  5  5  4 
+++ 1 ++ 5 ++
++ ++
...because the doublefive is the spinner, and not the doubleone. Got it?
So when can I play off of the spinner?
Only when it has a tile on each side already. For example, legal:
++
 2 

 5 
++
++ 5 ++
 1  5  5  4 
++ 5 ++
++
...illegal:
++
 2 

 5 
++
++ 5 
 1  5 
++ 5 
++
Also note that the illegal one just looks dumb. Don't branch until the spinner is "capped". Done.
So how do I get points (to win)?
This is the only tricky part, and its only tricky if you can't add. You can only score if the values at the ends total to a multiple of five. Hence the name. This rule is nonnegotiable, and does not change with the size of the set of dominoes. Doublesix, doublenine, doubletwelve and doublesixteen all can add together to get multiples of fives. Plus, multiples of five are the easiest to spot (they end in a 5 or a 0). So deal with it.
What do you mean by "at the ends" though?
I knew you'd have trouble with that, but it's best learned through example. Let's present the following situation:
++
++ 5 ++
 4  5  5  1 
++ 5 ++
++
The values in this situation are the 4 on the left end (not the five its attached to) and the 1 on the right end (again, not the five its attached to). Luckily for the last person to put down a tile, they scored FIVE points, because the sum of the ends is FIVE (4+1=5, remember?), and FIVE is a multiple of five.
Here's an example with branching in effect:
++
 2 

 1 
++ ++
+++ 1 ++ 5 ++
 3  6  6  1  1  5  5  4 
+++ 1 ++ 5 ++
++ ++
Here, the 3 on the left, the 4 on the right, and the 2(!) on the top are on the ends. Note that the 1 on the opposite end of the spinner does not count as an end, because nothing has been played on it yet. Note that since the total of the ends is 9 (3+2+4), the person who played this receives no points, because 9 is not a multiple of five.
Another example, this time with a fun situation:
++
 2 

 1 
++ ++
+++ 1 ++ 5 +
 3  6  6  1  1  5 
+++ 1 ++ 5 +
++ ++
Here the values at the ends are the 3 on the left, the 2 on the top, and BOTH 5's on the right. This gives a grand total of 15, which is a multiple of five, and is thus 15 points to whoever just played.
With this property of doubles being established, you may have thought of something. "Isn't starting off with doublefives an instant 10 points then?" That's right. However, if your opponent plays a fiveblank off of that, that gives them 10 points as well (0+5+5). Also, if two doubles are left uncapped, and branching is in effect, the opportunity for huge scores is a viable opportunity. Tread with caution.
That's the only way I can score?
No, it's not. If you happen to win a hand (by being the first to run out of tiles, remember?) you get the sum of all of your opponents tiles. Of course, to keep with the multiple of fives theme, the total is rounded to the nearest multiple. That is, if the total is 27, it is rounded to 25. If it is 28, it is rounded to 30. And so on. Thus, it can be quite profitable to win a hand, and it is good strategy to get rid of hefty tiles early, lest you be stuck with them when the hand is over.
Anything else?
Yes. Scorekeeping. You can do it as you like, just adding numbers like you would in Scrabble is good enough. However, since it's all in multiples of five, a shorthand system can simplify the process. The version I learned is reportedly how score is kept when playing this game in the California prison system.
The basics for the shorthand system are that a slash signifies 5 points, and a circle signifies 10 points. The first two slashes should be made to form a large "+", like an xy axis. Each quadrant of the + should be filled with either a circle or an "X", composed of two slashes. With the four quadrants each being 10 points (either an "O" or an "X"), and the cross itself being 10, the completed axis and quadrants signifies 50 points. Easy, right? After a 50point unit is filled, start another one. Using this system, it's easy to calculate your current point totals, and constantly tally up the score. Plus, since the goal points for victory are usually in multiples of 50, it can be measured in "units" instead of continually added numbers. Here's an example:
Brian  Tom
+
O  X  X  X
+  +
X  O  O  \

 
 
 
Clearly, Brian has 55 points, and Tom has 45. Remember, the slash is 5 points.
Any closing words?
How kind of you to ask, as that segues nicely to my closing statement: While the core rules of the game are hard and fast, feel free to experiment. Play with way more people than is optimal. Play up to 1000 points (if you want to be there for a while). Make it a drinking game (for instance, every five points scored is a drink to be distributed). Go nuts. And have fun.