If you need official confirmation, you can check EO Select Check, which contains the data from Publication 78 at the IRS website online. That publication lists most of the organizations that qualify as tax exempt. To get another copy of an exemption letter, write to the IRS.
If you find the IRS website intimidating or less than user friendly, you can also get confirmation by going to www.guidestar.com. Guidestar has information and Form 990 tax information returns on most charities in the country. If you have been filing annual tax returns (other than the 990-N), they will probably be shown on the Guidestar website.  If you haven't been filing any returns at all, you may have lost your exempt status.  The Pension Protection Act of 2006 provides that most exempt organizations that fail to file a required tax return for three years in a row automatically lose their exempt status.  The first list of groups that have lost their exempt status should be released sometime in 2011.

2023 Update:

Since this question was originally answered, the IRS has significantly upgraded its website and the capacity of individuals to check on the status of exempt organizations.  Check out the Tax Exempt Organization Search (TEOS) feature on the IRS websie at IRS.gov.  You can search on an organization by name or by EIN.  You can normally review their recent tax returns, and download determination letters issued since January 1, 2014.  If you need copies of other documents that are available from the files, you can request copies using a revised Form 4506-A for a tax return not available on the website or 4506-B for a copy of the application for exemption or a determination letter not on the website.  —Don Kramer