This page is just meant to be a quick and dirty overview of the different types of Go rules one might encounter out there. For more detailed descriptions and explanations you can check one of the following pages:
Anyway, here we go...
Rules of Go in all their glorious variations can basically be divided into two categories:
The main difference between both these types is the scoring method. In both cases the scoring is performed by calculating the number of points each player earned, and then comparing these two numbers against each other. Whoever has the higher number (called the higher score), is the winner. So far the similarities. The differences between the two scoring methods are in determining the score for each player.
The japanese-style rules are by far the more popular in the western world, so lets start with them.
The most distinguished feature of japanese-style rules it the scoring method, called territory scoring. It is performed by counting the number of empty intersection each player surrounded and adding to this number(s) the amount of enemy stones each of the players captured. In a philosophical sense - this makes Go a game of surrounding because your score is determined by what you managed to surround (which includes empty intersections and captured stones.)
In other words, the score for each player is determined by the following formula:
Player's Score
= (number of surrounded empty intersections) + (number of captured stones)
This is pretty much the rules and the scoring method I use to teach Go, and this is what you will find on most of my Go Rules pages.
Although not quite as popular as the japanese-style rules in the western world, the chinese-style rules are slowly gaining a secure foothold. The reason is that they seem to circumvent some of the pitfalls inherent in japanese-style rules.
The main distinguishing feature of chinese-style rules is their scoring method, called area scoring. Using this method, the score for each player is determined by adding the number of empty intersections enclosed by this player and adding it to the number of stones the player has on the board. In a philosophical sense - this makes Go a game of dividing because the game is decided by who managed to cut a larger slice of the board (if you imagine the board being divided into two parts - the ones controlled respectively by each player and including both the empty intersections surrounded by the stones and the surrounding stones themselves.)
In other words, the score for each player is determined by the following formula:
Player's Score
= (number of surrounded empty intersections) + (number of stones of corresponding color on the board)
Of course, the two main rule-styles have other features than their respective scoring methods, and thus other differences. These differences range from the ways of treating specific positions, ways of deciding which points are to be counted and which not, and even which games are indecisive by definition and what happens in such cases when a game cannot be decided.
One of the main differences between rule sets (apart from their scoring methods) is how they handle a situation called ko (explained here), in particular if they use a "superko" rule or not (basically, the superko is a rule which prohibits global whole-board position repetition, while simple ko rule only deals with immediate local position.)
There is pretty much it as far as rules are concerned. Most rule sets used in practice represent some kind of compromise between the two above styles. Sometimes special provisions (like Pass Stones) are supported to wash out differences between the rules styles. Sometimes the efforts to create a perfect rule set can be quite esoteric or even impractical, but still, if you understand the two main rule styles I described above, you are well equipped to be a force in the world of Go. :)
The main practical difference between the territory scoring method and area scoring method is that under territory scoring you lose one point if you play inside your won territory, while under area scoring you do not. After all, area scoring counts both the empty intersection you surrounded and the stones you have on the board, os if you play inside your own area - you decrease the surrounded area by one, but at the same time you increase the number of stones you have on the board by one - so the final score remains unchanged.
Under territory scoring, playing inside one's own territory, if not really necessary, loses you a point. Consider - territory scoring counts the empty intersection you surrounded and the enemy stones you captured. By playing inside your own territory you decrease the number of empty intersections you surrounded by one, but you do not increase the number of enemy stones you captured, so your score is changed - you lost one point.
As a result, you might safely play inside your own area under area scoring while under territory scoring you better think hard before doing that, consider if it is really really necessary - or you might lose a precious point.
However - the butter is not all on one side. Under territory scoring, the decision about passing or filling in a dame (neutral) point is painless, while under area scoring passing instead of playing a move in neutral area costs you a point because by playing another move on the board which is not inside your own area you actually gain a point (and so you fail to gain one by passing.) This basically exchanges one kind of tactical decision (do I need to defend inside my own territory?) for another (can I pass or do I have to play in neutral area?) Some say that the extra level of complexity introduced by the territory scoring is more interesting that the one under area scoring, and they favor territory scoring. I belong to these people.
You can find an example of territory scoring method applied to an example position here.
On the other hand, the area scoring method is an excellent solution to a problem mentioned in the page about end game and lack of agreement between players (see here.) The very first situation I describe on that page is when players disagree about the status of a group. One of the possible solutions I suggested was to build the position on another board and play it out to decide the status. Now, you might wonder why I say that the position is to be created on a separate board and not just played out on the one in use. This is because under the territory scoring method further play might affect the outcome of the game - i.e. one of the players might make unnecessary move(s) which will cost him/her to lose points.
Area scoring method allows for the position to be played out safely, without affecting the final score. After all, playing inside your own area does not affect the outcome, and since enemy prisoners are not counted, any extra stones you might lose in the process are also meaningless.
This actually encourages people to play things out, especially beginners, which can help to dispel the confusion and difficulties surrounding potential questions about the status of groups. And this is one reason why many people prefer the area scoring method. I myself have to admit that it seems more suitable for teaching the game and for beginners to use in their own games.
However, as already mentioned, territory scoring is by far more popular in the western world. So there seems little point to teach beginners the "superior" area scoring just to have them go to a club and get confused by another method. Or, at least, this is what I think.
You can find an example of applying area scoring method to an example position here.