Author Existence Failure: Something that fans continually fear, especially since Martin is well into his sixties and it took him nearly six years to complete A Dance with Dragons. What also doesn't help is that he has gone on record to state that if he does die, he won't allow another author to finish his work. Fortunately, he already knows how the series as a whole will end, and has told the producers of the Game of Thrones tv series several major plot points in advance.
"Un-Cat" or "Zombie Cat" for Catelyn after she's resurrected.
In certain parts of the community Ser Gregor is known as "The Mountain that Rapes". You can guess why. Later on, i.e. as of A Feast for Crows, Qyburn's experiment with Gregor was dubbed "Gregorstein", "Qyborg" or just "Un-Gregor." In A Dance With Dragons it's given an official name: Ser Robert Strong.
"Raven Email" (also adopted by viewers of Game of Thrones), in reference to the oddly fast and reliable method of communication via carrier-raven. Similarly, the use of weirwood trees to see the past and present is often referred to as the Weirwood internet or weirwood.net (and more advanced Greenseer abilities are naturally the weirless internet). An alternative fanmade name for the Weirwood internet is the Tree Matrix, since Greenseers have to literally plug themselves in.
In A Dance With Dragons, Manderly's revenge scheme is generally referred to as Delicious Frey Pie, with means exactly what it sounds like.
Littlefinger is frequently abbreviated to simply "LF". Or people call him Creepfinger.
"Robbwind" or "Grey Robb", for the desecrated corpse sewn together from Robb and his direwolf's remains.
Stannis is affectionately known as "Stannis the Mannis" by his fans, often shortened to "Stan the Man". Also, "Stanislav" which is a portmanteau of sorts for "Stan is love", itself a reference to a meme.
The Other that Sam kills in A Storm of Swords has become known affectionately as "Puddles". Due to the fact that the Others seem to "melt" when killed.
fAegon, as in Fake Aegon for people who don't think Young Griff really is the son of Rhaegar. Jeyne Poole is sometimes similarly called fArya because she was forced to impersonate Arya Stark.
In some circles, Jon Snow's murder is referred to as "The Ides of Marsh", because he was killed by Bowen Marsh and several of his co-conspirators in a scenario similar to the assassination of Julius Caesar.
Melisandre often gets shortened to Mel.
Jorah's unrequited feelings for Daenerys earned him the nickname "Ser Friendzone".
Based on correspondence with GRRM, who refused to confirm that Rhaegar's son Aegon was killed during the Sack of King's Landing, some fans began to believe that it was a decoy that the Mountain snatched from Elia's hands and smashed against a wall. Guess who shows up in Tyrion's chapters?
Also, in DVD Commentary for the TV show, GRRM confirmed that the undergaoler "Rugen" is one of Varys's disguises.
Saved from Development Hell: A Feast for Crows and A Dance with Dragons both took years to write, causing some fan complaints to reach a fever-pitch, but both were eventually completed.
Shrug of God: After the release fiasco that was A Dance with Dragons, Martin no longer risks giving a concrete estimate for when his next book will be finished. He has also remained frustratingly vague on whether he will wrap up the series in the next two books or if he will expand it to three or more.
Vaporware: A Dance With Dragons was "forthcoming" in one way or another starting in 2004, but was published at last on July 12th, 2011. Time will only tell if The Winds of Winter and A Dream of Spring will eventually fall under this trope.
Martin originally planned to have a five-year Time Skip between the third and fourth books, which would have had a major effect especially on the several child and teenage characters. In the end, he wasn't able to pull it off. And ironically, there actually was a five year gap between the two books' publication. He lampshades it with one character saying (paraphrased) that he "expected five years of peace, at least, before Cersei screwed everything up."
Several chapters were edited out of A Dance With Dragons, including ones for Aeron Greyjoy and Sansa, who didn't have chapters in the finished version. Their chapters have reportedly been moved into The Winds of Winter, so we'll have to wait and see if that is the case.
The Wiki Rule: A Wiki Of Ice And Fire is the biggest, and can be found here. There's also A Song Of Ice And Fire Wiki, which can be found here.