On 2nd April 1792, the U.S. Congress passed An act establishing a mint, and regulating the Coins of the United States, more commonly known as The Coinage Act. This legislation established the United States Mint at the seat of government, then Philedelphia, and several official positions within the mint to conduct its business: a Director, an Assayer, a Chief Coiner, an Engraver, and a Treasurer. The act allowed for the roles of Chief Coiner and Engraver to be filled by one person.
The act also established a decimal system of currency for the country and set the various denominations of the coins. The three highest value coins were all made from gold: $10 Eagles, $5 Half Eagles, and $2.50 Quarter Eagles. Next came five silver coins: $1 Dollars or Units, $0.50 Half Dollars, $0.25 Quarter Dollars, $0.10 Dismes, and $0.05 Half Dismes. The $0.01 Cent and $0.005 Half Cent were to be made from copper.
Section 10 described the legends and devices to be stamped on the coins:
Upon one side of each of the said coins there shall be an impression emblematic of liberty, with an inscription of the word Liberty, and the year of the coinage; and upon the reverse of each of the gold and silver coins there shall be the figure or representation of an eagle, with this inscription, "UNITED STATES OF AMERICA" and upon the reverse of each of the copper coins, there shall be an inscription which shall express the denomination of the peace, namely, cent or half cent, as the case may require.
According to the legislation, anyone could bring gold or silver bullion to the mint and have it made into coins free of charge or immediatly exchange bullion for the equivalent weight in coins. If you wish to read the full text of the act it is available on the Library of Congress web site.