"Malfoy called Hermione something -- it must've been really bad, because everyone went wild." 
     -- Harry Potter

taunts and insults
"potty wee Potter"
Peeves calls Harry (CS11)
"Potty and the Weasel"
Malfoy calls Harry and Ron (
"mudblood"
extremely insulting: "not of a pure Wizard family" (literally: "dirty blood") (CS7)
"the famous Harry Potter and his faithful sidekick Weasley" 
Snape calls Harry and Ron (
"squib"
non-wizard born of a wizard family (CS9)
"Sir Properly-Decapitated Podmore"
Nearly Headless Nick sarcastically calls Sir Patrick Delany-Podmore this (CS8)
"Scarhead"
Draco calls Harry during a Quidditch match (
unrecorded nasty names
Ron calls McGonagall these (PA8)
making faces
grindylow (PA8)
"pea-brain"
What Ron called the Mountain Troll in the girls' bathroom. (SS10)
"lousy, biased scumbag"
What Ron called Karkaroff when he gave Harry a mere 4 for a score on the first task (GF26)
"Pretty Boy Diggory"
Seamus Finnigan called Cedric this, and the Huflepuffs glared at him (GF16)
"dung brains"
Fred calls Ron this (GF16)
"long-haired pillock"
Rita Skeeter called Bill Weasley this in an article. It didn't seem to bother him much. (GF10)

nicknames
Mad-Eye (Alastor Moody)
old disfigured Auror who has one magical eye which moves around uncannily (GF)
Padfoot (Sirius Black)
Marauders nickname of Animagus who can turn himself into a large dog (PA, GF)
Prongs (James Potter)
Marauders nickname of Animagus who can turn himself into a stag (PA)
Moony (Remus Lupin)
Werewolf, gained this nickname as part of Sirius and James' group, the Marauders
Wormtail (Peter Pettigrew)
Marauders nickname of Animagus who can turn himself into a rat (PA, GF)
Your Wheezy
Dobby refers to Ron as Harry's Wheezy (GF
Snuffles
Sirius asked Harry, Ron, and Hermione to refer to him as this (GF
my sweet
Filch calls Mrs. Norris this 

fun things to call Percy
Bighead Boy (play on "Head Boy") (PA4)
Humongous Bighead (from the HB badge the Head Boy wears) (PA4)
Weatherby (Crouch, his boss, called him this)
Perfect Percy (play on "Prefect")
Pinhead - George and Fred bewitched Percy's prefect badge to say this (CS)