Overview
Internet Slang is as old as Internet itself and has spawned a huge amount of different ways of speaking, sub-languages, expressions and idioms that have got the most of their meaning on the Internet. Those different kinds of language can be either known as chatspeak, SMS speak or IM language. Appreciated by some, despised by others, they nevertheless have taken hold as a strong element pertaining to the web since their birth to these days.
Intentional Misspellings
The heart of Internet Slang is to be based on a modified version of our current alphabetical way of speaking. Because of space restriction existing since the dawn of Instant Messaging, SMS and services such as Twitter or because of a will to gain time at the expense of understanble grammar in sending a message, explaining why a big part of the web tends to hate the ones who use it, those kinds of misspellings became words on their own.
Phonetic Translation
Coming straight from SMS language, this is a way of replacing entire words or bits of words by their phonetic equivalent embodied into single letters or figures.

A few examples below :
2 => to
u => you
gr8 = great
4 => for
da => the
y => why
Many memes tend to reuse them on their own, as can be seen by examples such as In Ur base or LOLcats.
A Russian version from that type of language exists as well, and is called Padonkaffsky jargon
Disemvoweling
Disemvoweling is the way of removing all vowels from a token word. According to wikipedia, the name exists since 1990 and it works as follows :
The quick brown fox jumps over the lazy dog
would, after being disemvowelled, look like this:
Th qck brwn fx jmps vr th lzy dg
Disemvoweling is also abused in SMS/IM writings as the above picture shows.
Leetspeak
1337speak consists of replacing a letter by its closest figure’s look-alike, or by using various keyboard signs such as “+”, “(”, “[”, “/”, “_” and more to recreate that letter.

A more detailed explanation of that phenomenon can be read in the 1337speak article.
A Filipino equivalent has emerged in 2010 : Jejemon.

Letter repetition
More than a slang, it symbolizes a type of language, often pointed as being used by teenagers, consisting of the same letter in a word repeated several times for emphasis, like the “gurlllll” word in the picture below.

Derived from it, other phenomena took off such as !!!111oneeleven as an ironic way to replace an exclamation mark or the FUUUUUU slang to express great frustration, and coming directly from Rageguy.
Acronyms and Initialism

Acronyms or the art of creating a neology through the association of letters and initials between them exist on the Internet since the late 1980s.
Starting with LOL, from laughing out loud and used to convey great hilarity, hundreds of others have been made since then. Among them can be found : OMG, BRB, RTFM or also tl;dr.
Idioms

More than pieces of languages, Internet is also the cradle of many expressions and usages which complete meaning can only be acknowledge and understood because of or in relation to the Internet.
While some of them are used as synonymous of another word, the way Over 9000 can be viewed as a replacement in expressing lots or bucketload the same way win and fail are respectively meaning success and failure, others have their own meaning, such as DERP as a way to signify stupidity, or also Fag as a suffix which has a totally different meaning and doesn’t necessarily address homosexuals in a derogatory way.