The first place you should go if you have a stock certificate that you would like to track is the Internet. Go to a major multi-search engine such as Google or Dogpile and see what comes up when you type in the name of the company. You can also go to any stock site and type in the ticker symbol if you know what it is (or was). This may or may not bring up any information on the company, but could at least tell you if the company still exists or if its name has changed.
If none of the free websites can help you, try stocksearchintl.com. This site will research any stock certificate that you input for an $85 fee. (The fee drops to $40 if the stock is already in its database.) If you're willing to pay an additional fee, the site can also help to collect any money that you're due. You can also try OldCompany.com for a report on your company and how you can contact any corporate successor that has since arisen.
If neither of the previous measures help, you can go to your stockbroker and deposit the certificate into your account there. Once the stock is held in street name, you may be able to find out the company's current status. Your broker can track the stock via the CUSIP number that is printed on the back--and can place a trade to sell the stock if it has any value.
This process may take a few weeks,and it may also take some time to find out when the stock was purchased and the purchase price. (You will need this information in order to compute basis when you report the gain or loss.)