The Mint made "presentation pieces" from its earliest days (1790s). They made "proof" coins (as we know them) in small numbers for collectors in the 1830s, '40s and '50s. When coin collecting became popular in 1858 (after the Large Cents were discontinued and people started collecting them by date) the Mint started producing Proof Sets for sale to collectors. Proof coin production was discontinued after 1916 and resumed in 1936. World War II interupted proof coin production in 1942. The Mint resumed proof coins in 1950. When clad coins were introduced in 1965 proof coin production was discontinued for 3 years (1965 - 1967). Special Mint Sets (SMS) replaced proof sets during those years. Proof coin production resumed in 1968.
There is some debate regarding precisely what conditions are required for a coin to have been struck as a "Proof". A number of opinions go with the 1830's as the earliest that true Proof coins were produced in the U.S. However, I have seen population reports and/or coins as old as approximately 1818 which have been certified as Proof by the major grading companies. I have seen quite a few more coins which I would consider Proofs, from approximately 1829, on, however, and more, still, dated from the 1830's forward. There are a good number of coins in Proof holders which do not necessarily stand on their own merits.
The answers so far have been regarding US coins. But the first Proof coins were produced by Great Britain in 1652. And the definition of a Proof as I know it is a coin struck on specially prepared planchets by specially prepared dies and struck more than once.
