Originally a counter-reaction to the traditional hot rod, a label recently applied to undriven cars and super high priced "customs". The rat rod's beginning was a throwback to the hot rods of the earlier days of hot-rodding, built to the best of the owner's abilities and meant to be driven. Rat rods are meant to loosely imitate in form and function, the "traditional" hot rods of the era. Biker, greaser, rockabilly, and punk culture is often credited as influence that shapes rat-rodding.
The typical rat rod is an early 1930s through 1950s coupe or roadster. Early (pre-World War II) vehicles often have their fenders, hoods, running boards, and bumpers removed. The bodies are frequently channeled over the frame, and sectioned, or the roofs chopped for a lower profile. Later post-war vehicles are rarely constructed without fenders and are often customized in the fashion of Kustoms, leadsleds, and lowriders. Maltese crosses, skulls, and other accessories are often added. Chopped tops, shaved trim, grills, tail lights, and other miscellaneous body parts are swapped between makes and models. Most, if not all, of the work and engineering is done by the owner of the vehicle.
Recently, the term "rat rod" (or rat car, as modern cars are not actually hot rods like the name suggests) has been used to describe almost any vehicle that appears unfinished or is built simply to be driven.
The origins of the term and cars is most interesting. Again, from Wikipedia:
The December 1972 issue of Rod & Custom Magazine was dedicated to the beater, a low-budget alternative to the over-polished, slickly-painted, customized early car. The beater could easily be considered a progenitor of the rat rod with its cheap upholstery, primer instead of paint, and lack of chrome or polished metals. However, owners of these beaters often had a high-dollar machine sitting in their garage.
As with many cultural terms, there are disputes over the origin of the term "rat rod". Some say it first appeared in an article written in Hot Rod Magazine by Gray Baskerville about cars that still sported a coat of primer. Some claim that the first rat rod was owned by artist Robert Williams who had a '32 Ford Roadster that was painted in primer. Hot Rod magazine has since verified this.Although the term likely started out as derogatory or pejorative (and is still used in this way by many), members of the subcultures that build and enjoy these cars have adopted the term in a positive light.