To understand plate varieties, you need to have a basic understanding of how engraved stamps are produced.
An engraver works with a die to engrave a stamp design in reverse. Dies are usually a single subject of the stamp design. The die is soft so that the engraver can work more easily. Once the engraver is done, the die is hardened.
The transfer roll is then created. Its job is to transfer the stamp design from the die to the printing plate. A transfer press rocks the soft steel of the transfer roll back and forth on the die. This presses the soft steel into every line of the die. A transfer roll typically contains several impressions of the die. Each impression is called a relief. Once all of the transfers of the die to the transfer roll are complete, the transfer roll is hardened.
The transfer press is used once again to transfer the stamp designs from the transfer roll to a flat printing plate. The printing plate may have tiny dots or lines on it to help align the transfer roll properly. The transfer roll applies the stamp design to the printing plate. After all of the stamps are created on the printing plate, the layout dots, layout lines, and any other markings not needed on the printing plate are removed. The plate number and any other markings (guide lines, etc) are added to the plate. The plate is then hardened and ready to print stamps. In the case of a rotary press, the flat printing plate is curved slightly before being hardened so that the printing plate matches the curvature of the printing press. (As a side note, this is why the stamp design of a rotary press stamp is always slightly larger in one direction than its flat plate counterpart).
Plate varieties can occur at any point in this three step process.
Die varieties occur in every plate position made by that die. These are caused by an imperfection in the die or an error by the engraver.
A relief break occurs when a tiny piece of the transfer roll breaks off, leaving an unintended white space on the printing plate. Every plate position made by the same relief will show the same plate variety. Relief breaks tend to progress over time from tiny to larger unprinted areas.
A plate variety in the printing plate comes in lots of different forms. Layout dots or lines may not have been completely removed. Dropped, double, triple, short, twisted, or shifted transfers occur as the transfer roll comes into contact incorrectly with the printing plate.
Plate varieties occur during plate use too. The printing plate may develop cracks or other damage due to wear and use.
Plate varieties generally fall into two categories. Foreign material may have fallen onto the die, transfer roll, or printing plate and became part of the final stamp design. These show ink where there shouldn't be any. Areas of the die, transfer roll, or printing plate become damaged or break away. These show blank space where there should be ink.
There are thousands and thousands of plate varieties. A few are recognized by the Scott catalog. The "broken hat" variety on Scott #231, the "broken circle" plate flaw on Scott #616, "gripper cracks" (where the printing plate is gripped to the printing press) are known on many issues from the 1920s and 1930s, and so forth.
Popular plate varieties like the broken hat on Scott #231 and other more striking varieties like large plate cracks have premium values. Many varieties though are tiny and often bring little or no premium value. The thrill is in the hunt!
Sometimes plate varieties are corrected. Stamps are re-engraved or retouched to eliminate the problem. The die, transfer roll, or printing plate is softened to allow the rework to occur. If the rework is done by an engraving tool, it's referred to as a re-cut or re-engraved stamp. If the work is done by etching, it's referred to as a retouched stamp. In general, a re-engraved stamp corrects a prominent problem. A retouched stamp corrects a tiny problem.
Loran C. "Cloudy" French wrote the "Encyclopedia of Plate Varieties on U.S. Bureau Printed Postage Stamps" in 1979. It is dated, but it is still the bible for plate variety collectors. There are other sources of information on plate varieties such as the Shift Hunter Letters published from 1929-1939.
Not every flaw you find on a stamp is a plate variety. Sometimes foreign matter may fall on the printing plate and create what looks like a plate variety. Although these items are collectible too, they are not plate varieties. The key to being a plate variety is that the feature must be constant.
Plate varieties are relatively common in the late 19th and early 20th century issues. Stamps were produced in large quantities. Printing plate production techniques were immature. To control costs, it was cheaper to overlook tiny flaws like relief breaks. Also, it was usually cheaper to retouch or re-engrave a stamp than it was to start making a whole new printing plate. By the 1940s, plate production processes improved to the point where there are very few plate varieties known.
No pun intended, but this information just scratches the surface of what's possible when you collect plate varieties. It takes a sharp eye and a good magnifying glass. And a lot of patience! Collecting plate varieties can be very challenging. And the good thing is that many varieties fall on the most common of stamps, such as the 1¢ and 2¢ Washington-Franklin issues. It doesn't take a lot of money to collect these kinds of stamps.