The 1881 three cent nickel (which has exactly the same diameter as a dime - 17.9mm) is a common date for the type. The coin is worth $15-$25 in circulated condition, $40-$60 in very nice circulated condition, and $100 or more in uncirculated condition. You should see a reputable coin dealer to get a better estimate of value.
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The coin is an 1866 3 cent piece not a dime, and it's made of nickel not silver. Circulated coins are valued at $15.00-$50.00 depending on condition.
It's not a dime, the III is the Roman numeral 3 so you have a 3-cent piece. That odd denomination was issued (along with a $3 coin) to make it easier to buy postage stamps when they cost 3¢. There are 2 varieties from that year: One variety is made of copper-nickel and has Miss Liberty's picture on the front. As of 12/2008 its retail value ranges from $15 in worn condition to about $30 with only slight wear. The other variety is made of silver and copper, and has a star on the front. It's much scarcer and is worth $335 to over $500 depending on its condition.
This is not a Dime. It's a 3 cent-piece. 1865 is the first year (1865-1889) of issue. Circulated examples have current retail values from $15.00-$70.00 depending on the grade of the coin.
Most of the silver coinage, like the half dime (worth 5 cents and struck in silver, the nickel wouldn't be minted until much later), the dime, the quarter, and the half-dollar used the seated Liberty design along with the silver dollar. The penny was a lot larger than the penny we use today, closer to the size of a half dollar, and used the braided hair design. There was also a three-cent piece which was really small, made out of silver and had a star on one side and a C and III on the other. There was also a small gold dollar, smaller than a dime but made out of 90% pure gold. There were other gold coins too, the quarter eagle, worth $2.50, half eagle worth $5, eagle worth $10, along with the double eagle worth $20, each of them struck with 90% pure gold (alloyed with 10% copper to make the coins scratch and wear resistant).
The III is its value. Its called a three cent nickel because its face value is three cents and its primary composition is Nickel. Depending on its condition it usually sells for $15 - $25. If its uncirculated it can sell in the hundreds of dollars.