For a good half-century now, the hobby of hot-rodding typically meant taking a cheap car, taking out any body part that didnt matter (i.e. roofs, hoods, bumpers, fenders, seats, and other such nonsense), modifying the engine and/or dropping in a bigger one for greater performance (often protruding upwards from the hood), and fattening up the tires for extra traction.
The term is still as accurate as ever. In fact, not even the cars in question have necessarily changed: one very typical image of a Hot Rod is a Muscle Car straight from the 1960s (the so-called muscle car golden age), restored to all its glory and then some. Its not uncommon to take the great ancestors of cars we know today (Mustang, GTO) or ones forgotten by all but a few (Plymouth Barracuda), and send the output of its V8 soaring to 600 horsepower and above. Hot rods can be just as much about customizing as weight-saving (think of flaming paintjobs), and price isnt necessarily an object: one notable Barracuda (Hemi Cuda in hot rod speak) on the cover of a major-name Hot Rod Magazine had every body panel and interior item customized to its owners desire. For $340,000.
As for hot rods relation...