By definition, a "chopper" is a motorcycle that has been stripped of every unnecessary item, making it as close to two wheels, a transmission, chain, handlebars, engine and controls as possible.
Back in the day, motorycles were not as high powered as the large, high-horsepower engines of today. A Harley Davidson "panhead" topped out at about 1200cc. So in order to try and eke as much speed out of their vehicles as possible, some people removed the heavy dual seats and replaced them with slim bicycle style seats. They jettisoned the high-weight touring bike front end with the wide tire, and replaced it with the much lighter front end off the 880cc, narrow tired Sportster models.
Wiring was reduced to the essential seven wires or so required to make a motorcycle actually run.
At the same time, the motorcycle was adjusted to make it more stable at speed, by moving the front wheel forward and out by lengthening the front fork tubes and changing the angle of the neck to adjust the front wheel outwards. Initial "engineering" attempts had no idea what "rake and trail" are, and as a result some of the initial experiments produced motorcycles that either handled terribly at low speed but acceptably at very high speed - or worse, extremely well at low speed and not at all at high speeds. (Rake and trail are measurements pertaining to the difference between a line drawn perpendicular to the ground at the front axle, and a line projected from the end of the forks, and the difference between a line projected from the end of the forks, and a similar line drawn from the neck.) A variety of engineering adjustments were used to correct these problems, including raked trees, which fixed a rake problem, and things like Sugar Bear's practice of putting the axle on the end of a curved bar connected to where the end of the fork would be, modifying the trail.
The practice of using a "tweek" bar, namely a connecting bar between the two fork tubes further down a particularly long set of forks, made the assembly less likely to "flex" under load and corrected some issues with length.
The practical upshot of this was that you'd have a motorcycle that stood "taller" at the neck, and sloped long and gently out from that neck to a wheel, sometimes ridiculously far out in front of the machine. Making turns, or worse, a U-turn was difficult and in some extreme instances almost impossible with these machines.
The chopper was usually also accessorized with Z-bars, bunny ears (a kind of curved handlebar) or "ape hangers" - namely tall handlebars desighed to put the rider's hands at shoulder height or higher. Forward controls were also popularly paired with these - making the rider able to "sit back" and stretch out both hands and feet for a far more comfortable distance ride.
There were some chopper builders back in the 1960s, but the real pride was in making your own - taking a metal saw to your frame and then heating the metal and bending the neck outwards, stripping the bike to the frame and only adding half the parts back, and chroming anything that could be chromed. Replacing tanks with a single, tiny "peanut" tank with a groovy metalflake or psychedelic paint job. Coming up with a bike that was customized to your personal aesthetic and taste.
Of course, it was also associated with the outlaw motorcycle club element, so pretty soon there were laws-a-plenty saying you could not have ape hangers, etc. etc. etc. making as many anti-chopper laws as possible. These are still in the books in most places, but usually ignored by law enforcement because of "rule of cool".
In the 2000s there was a revival of chopper culture, only this one was far less about shade tree mechanics and more about building up custom bikes from parts. You did have your Indian Larrys and Billy Lanes and Jesse Jameses producing painstaking metalwork the old school way, but the new style was buy parts, bolt them together, and bling it all out with as much billet aluminum as possible. Suddenly the market was flooded with Exile Cycles, Big Dog, Bourget, etc. etc. etc. all of whom were making high dollar machines by combining parts out of catalogs - the most famous of which was the soap opera reality television series American Chopper with the Teutul family.
That scene roared up and became HUGE, with lots of players and fortunes made, and just as quickly, fortunes lost. People spent tens of thousands on these machines and then suddenly they were near-worthless.
To tell the truth, most motorcycles are customized somewhat - anything you have that personal a relationship with, you're not going to leave standard. It's been a reason for Harley Davidson's success, starting with a base price but making sure you know all about the chrome accents, new pipes, special bits and pieces and custom color schemes they offer as upcharges.
But to some people, buying a motorcycle, taking it home and then taking it apart is half the fun of ownership.
To go in the other direction see dresser.