Occasionally when I send someone an e-mail, they will write back telling me that my message included a mysterious attachment named "signature.asc" which they can't open. I refer such people to this page for an explanation.
The "attachment" you are seeing is actually a digital signature. Just like a physical signature on a paper document, this digital signature certifies that I am the author of the message. Unlike signed paper documents, however, which are relatively easy to forge or otherwise tamper with, digital signatures can give you a very high degree of confidence that the message is genuine and unaltered.
Digital signatures have been a standard part of e-mail since 1996, and most e-mail clients handle them correctly. However, there are two notable exceptions: Microsoft Outlook and Microsoft Outlook Express. These two e-mail clients fail to interpret the signature correctly, unhelpfully displaying it as an attachment which cannot be opened.
If you are using Microsoft Outlook or Microsoft Outlook Express and have any choice in the matter, please consider switching to an e-mail client that isn't broken by design. (Not only do these two clients fail to handle digital signatures properly, but they also flout or fail to support a number of other important Internet standards and best practices. These shortcomings affect not only you, the user, but also often create problems and annoyances for the recipients of your messages.) Fortunately, there are a number of free e-mail clients to choose from, perhaps the most popular of which is Mozilla Thunderbird.