Rules As Written
Rules As Written means the exact words of the text, grammatically parsed for direct meaning. This means the text of rules is taken as a literal thing, even if that would otherwise break the 'physics' of the world or other things. Rules As Written is often used behind some of the silly jokes you'll see in D&D - such as the peasant rail gun which involves lining up 10 miles of peasants or more to pass a rock 10 miles in the 2-3 seconds of a turn and accelerate it to near relativistic speeds - and is also the key to defeating those zany scenarios - after all, RAW, a peasant throwing a rock does 1d4 +/- strength mod, regardless of how fast the rock is going when the peasant throws it.
Rules As Written is, generally speaking, an easier place to take defense because words have specific meaning and the text will directly support you. If the rule says a character can take no more than 2 attacks a round, then it doesn't matter if they have 8 ranks of extra attack because they can only take 2 attacks. Easy ruling, and easy to point to when making the ruling. However, RAW often gets into trouble from two simple facts: grammar makes meaning debatable in some sentences, and the writers/publishers are human and thus fallible.
For grammar making a sentence mean two different things I don't think I need to explain. All it takes is some vague wording and you can be wondering if you deal 10d6 damage AND healing to the target of a spell, or if you do 10d6 damage to the target and heal that amount as the caster.
Humans being fallible on the other hand can have some fun results. Consider for example the standard wording for adding bonus rolled dice to a check in Legend of the Five Rings 4th edition. "When making a
Rules As Intended
Rules As Intended is making ruling as the creator of the game intended the rule to work. Someone using Rules as Intended understands that you can't add Xk0 to an already made check, and thus have to add it to the dice pool used to get the result.
The problem with Rules as Intended is that you do have to interpret the intent of the rule, and sometimes that intention is pretty far from what the RAW actually says. Because of this you can also get into a lot of discussion over what different rulings mean, when they mean that, and when that means other things.
However, it can be the difference between understanding that a magical lock should have a key - and that key is owned by the creator - and a magic lock permanently sealing something in such a way no one can get to it ever again.
Both Is Best
Like with most things, using both is best. Sometimes parsing the wording of the rule is enough. Other times going with the intent behind the rules works. Even other times, using your own ruling to keep things going is even better.
Still, it's good to know how the terms work for those discussions, and to be ready to use them when explaining how and why you're making a ruling or deciding to handle a situation.