Grammar is the set of rules of language and how we use it. Correct syntax, sentence structure and punctuation are all part of grammar.
People who use correct grammar capitalize the first letter of their sentences and end sentences with a full stop. They use commas and apostrophes when they're supposed to. They write sentences which neither run on or are sentence fragments.
Apparently, being on the Internet is an automatic exemption from using grammar and spelling. It's true, ask anyone why they don't use grammar on the Internet and their reason is always "Because it's MSN/the Internet/IRC". There's no real reason to not use grammar. Grammar is out of fashion nowadays, you're more likely to be ridiculed for using correct grammar than not.
A little lesson on common grammar errors:
Your - Implies ownership, example: "Your dog" or "Your house", NOT "Your stupid".
You're - Used to describe someone when talking directly to them, usually followed by an adjective or "a/an (insert noun)", example: "You're stupid".
There - Referring to location, example: "He's over there" or "Are you there?".
Their - Implies group ownership, example: "Their house" or "Their rights".
They're - Contracted form of "they are", example: "They're quite stupid".
It's - Contracted form of "it is", example: "It's really hot outside", NOT "A cat and it's kittens".
Its - Implies non-human ownership, example: "A dog and its bone".
He's - Contracted form of "he is", example: "He's so sexy", NOT "He's hat".
His - Implies male ownership, example: "His hat", NOT "His so sexy".
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