Document IDs are a convenient way to track documents within a site collection. Previously, if you moved a document around (or renamed it or its parent library), links to the original will break. With Document IDs, you can still track the document around using a unique ID (site collection scope).
To use this feature, you need to activate the site collection feature Document ID Service first. Once activated, go to Document ID Settings (under Site Collection Administration, from site settings) and check the Assign Document IDs. Optionally, select a prefix and also force reset of IDs (you should not do this again under normal conditions).
From now on, every document you create (or edit, for those before service activation) will be assigned a new and unique ID. SharePoint uses a URL in the format http://site/_layouts/DocIdRedir.aspx?ID=DocumentID to locate the document. In fact, you can add a simple web part called Find by Document ID (from the search group) to a page and enter an exact ID, press a button and SharePoint will open the document right away!
You can, of course, surface the Document ID in a view, print it on a document label or even look it up in normal search.
This feature is not available in the Foundation version.