Generally those who have an affinity for cars and technology also have an affinity for performance, namely making cars go faster. The commonly discussed practice is to modify the engine to produce more power. There are also many suspension upgrades that can improve specific handling circumstances. Lastly there are cosmetic changes and audio modifications that can be made to enhance the image of the vehicle and/or owner. Please note this article is not a how-to, but rather a general introduction intended to inform those with little idea of what hot rodding is about.
The easiest and often most overlooked way to improve performance is to reduce the vehicle's weight. Which takes less energy to speed up to 60mph, a 2500 lb car or a 3500 lb car? Obviously it is not always easy to just shave off pounds (ask your Aunt Grezelda), and some vehicles will never be light without radical surgery, so it might be worth while to start off with a lightweight vehicle instead of a two and a half ton Cadillac.
There are a few considerations when choosing a hot rod. Japanese and British sports cars are known for their light weight, but can't always easily fit a V-8 engine, one of the staples of making big power. American muscle cars are known for being sold with V-8s, but are also considerably heavier. As an example, early '80s Mazda RX-7s weighed just over 2200 lbs and were sold with 100hp rotary engines, to put in a V-8 would take either major surgery or an expensive kit. Many early '80s Chevrolet Camaros weighed over 3200 lbs and were sold with 170hp 305 V-8s (some versions of this car were sold with larger and more powerful engines). Their stock performance is not very different, but with a Camaro one can easily swap in a direct replacement motor with over twice the horsepower.
If swapping in a new motor is out of the question there are always modifications that can be made to the existing motor to produce more power. Most common modifications are designed to help the engine 'breathe easier'. Internal combustion engines can be described as air pumps, the more air they can move into the intake and out the exhaust the more power they can make. Reducing airflow restrictions produces more power. There are aftermarket intake and exhaust manifolds that have larger and/or better designed passages, there are aftermarket camshafts that change how the intake and exhaust valves open, and there are methods of forced induction that cram more air through the motor. Another modification is enlarging the displacement (the swept volume within the cylinders) of the existing motor. This can be done by 'boring' the cylinders to a larger diameter, which requires new pistons, or by installing a crankshaft with a longer stroke, known as 'stroking'.
The suspension is often overlooked. Suspension does not affect a vehicle's acceleration or top speed as much as the motor does, but it has a very strong influence on the vehicle's ability to handle corners or irregular surfaces. Most vehicles are equipped with a loose, soft suspension that is more comfortable to passengers. Hot rodders are often willing to give up some of this comfort to gain performance. Thus, most suspension modifications center on tighter fitting bushings and firmer springs and shocks. Wider tires with shorter sidewalls also improve handling by means of a larger road contact patch and less flex in the tire.
Cosmetic changes are somewhat controversial in hot rodding circles. There are benefits to having a nice paint job, shiny chrome wheels, and a mega-watt sound system, but none affect the performance of the vehicle. Some have a minimalist approach to their hot rods, only painting it to keep rust from forming, keeping the stock wheels, and trying to make everything as maintenance-free as possible. On the far, far opposite end of the spectrum there are 'trailer queens'. A trailer queen is a vehicle built only for car shows. They usually are brimming with frivolous, overly picky details. From chrome plated brake rotors and chrome plated valvetrain that would be ruined if the vehicle were ever actually used, to twenty layer paint and custom one-off wheels, trailer queens are usually gawked at in wonder. They can be horribly expensive, but tend to have only aesthetic value. They are also the butt of many jokes relating to their lack of use.
Ultimately a hot rod is an extension of the builder, limited by imagination and available technology. They are an expression of creativity, desire, and power in one handy drivable package. They're also pretty damn fun!