Back in the days when coins carried actual value in precious metals like gold and silver, enterprising individuals, also known as criminals, would scrape a little metal off the edge of coins and collect a tidy stash of filings to be melted and sold later. Some circulating coins, therefore, were underweight.
To combat this thievery, mints would issue precision weights in non-precious metal (like brass) that could be used to check coins for proper value. Now these weights are highly collectible items.
If you do a search on 'coin weights' you will mostly find recent issues with dates 1821 and up. In fact, coin weights go back much, much farther, as evidenced by an amazing array of them at ABC Coins and Tokens in Alnwick, UK. You can scan ABC's list to get a good idea on pricing. As always, condition means a lot to value, so catalog values run like this:
worn: $20 US dollars approximate catalog value
average circulated: $40
well preserved: $80
fully uncirculated: $200
The coin weight in our picture comes from Munzhandlung Ritter in Dusseldorf, Germany, and CoinQuest thanks them for use of their image.
The numbers on British Royal Mint coin weights are intriguing. Chrisd asked about his (or her) coin weight dated 1843 with Queen Victoria's picture. If you look closely, you will find a small DW above the first number and a small GR above the second number. These are pennyweight and grains, respectively. Check wiki.answers.com for a thorough explanation.