It's a little more involved than simply banging someone's head against the curb. It actually means to force someone to lie down, open their mouth and bite down on the edge of the curb, then you stomp on the back of the person's head with as much force as possble(presumably breaking the person's lower jaw clean off or even cause death). I've heard the term used as far back as 1993.
You can see an example in the movie "American X".
by Mutex September 10, 2003
I am sure it existed before the song but on the album "The Incredible Shrinking Dickies" in 1979 there is a song called "Curb Job" as far as I remeber it was always a skinhead thing where a person bites the curb and has the back of there head stomped- and would be shown in the film American History X many many years later
"Curb Job"

Gimme a bone
I gotta get away
You better gimme a bone
There'e no way I'm gonna feed your face

Aaaoh
I'm gonna give you a curb job
Aaaoh
I'm gonna break your face
Aaaoh
I'm gonna give you a curb job
Aaaoh
I'm gonna break your face

Gimme a bone
I don't have one today
Gimme a bone
There's no way I'm gonna play your game
by Todd R. Emmerson June 29, 2005
Bang Somebody's Head against the curb.

NY street slang 90's
these suckas be getting kurb jobs
by LEc2 May 14, 2003
When you approach a car that is parallel parked and are fellated through the window.
She's such a crazy slut that she gave me a curb job at 4 p.m.
by AlaskaFister January 13, 2010
a russian curb job is when a person knocks out another person, puts their open mouth on the curb, with their teeth on the concrete, and kicks them in the back of the head, breaking many teeth.
(brains may come out, and head may flatten)
if you have an example, your sick...
Joe: so i gave this ass a russian curb job last week.
Devon: wow, dentures for life for that ass...
by aceponyboy October 26, 2007
The process of obtaining a “dumb hoe” in dark area. In order to get the “dumb hoe” to “eat your dick” call onto a “down ass homie” for assistance in “slam kicking” the “dumb hoes” neck in direction to land “dumb hoes” mouth onto your dick at a strong enough impact so that the “dumb hoe” has ability to “garble your balls deep in her throat”

*If successful you may see “dumb hoes” eyes pop out her head

Defetion provided by the dopest fort maker this side of the north bay ca
Damn John, thanks for helping with that comatose curb stomp blow job kick last week, that dumb hoe got deep past my balls with your help.
by LakeCountyTreatsYouRight February 9, 2018