Collectible coins can be categorized into various types, including commemorative coins, bullion coins, and numismatic coins. The United States has issued several collectible coins, including the Fugio cent, which was the first official coin of the U.S. minted in 1787. The four-dollar gold coins, known as the 1849 and 1850 Liberty Head designs, are rare and highly sought after by collectors due to their historical significance and limited mintage. Collectors often focus on factors such as condition, rarity, and historical context when evaluating the value of these coins.