Coins that are considered valuable are often coins that have some sort of defect, were produced in a limited quantity, or were produced far in the past and have been out of circulation for a long time. Additionally, coins that meet those criteria also fetch a higher amount if they have been preserved in good condition.
Some coins are rare simply because they are made of different material than usual. For instance, in 1943, the United States penny was made from steel, rather than the regular copper used in the years before and after. This also means that the few 1943 pennies that were made from copper are also rare and valuable.
Coins with defects are also valuable simply because they are outside of the norm and usually found in limited quantities. The 1982 U.S. dime minted in Philadelphia was accidentally sent into circulation without the small "P" that became part of the mintmark in 1980. Because this mistake only occurred at the one mint, these coins could fetch a high sum if sold.
A "double die" penny – a U.S. Penny engraved unevenly due to a production error – was accidentally produced in 1955. An estimated 24,000 of these pennies were produced.