The three cent piece (commonly referred to as a trime) was made from 1851-1889. The trime was made because the postal rate had just changed from five cents to three cents, and so this coin was an easy way to pay. The alloy was originally 75% silver, but in 1854, it was changed to 90% silver. The alloy was changed once more in 1865 to copper nickel, although circulation silver trimes were still minted until 1872. The change in alloy from silver to copper-nickel was to prevent the hoarding the trime and other silver coins went through during that time period. 

The trime is one of the few US coins to have the denomination in Roman Numerals (the only other coin I can think of off the top of my head is the Liberty Head Nickel, but I am sure there are others). It is the lightest coin the US has ever made, with it sometimes weighing 4/5 to 3/4 of a gram depending on the alloy. It also has a diameter of 14mm, making it the second smallest US coin to the gold dollar, which is 13mm in diameter, All trimes were minted at the Philadelphia mint, with the exception of 1851, in which trimes were struck at the New Orleans ("O") mint. The copper-nickel three cent piece is NOT a trime, but a three cent nickel. Please do not get it wrong. Proof specimens were made in 1858 and every year onward until the series ended in 1889. The key date of the silver specimens is the 1872, and the key date of the copper-nickel versions is the 1885.