What is this signature.asc attachment on your emails?
Most emails which I send are automatically digitally signed, so that it is possible to verify that it was actually me who sent the message, and that the message has not been altered since I sent it. The file attachment called signature.asc is the cryptographic digital signature itself, encapsulated in OpenPGP/MIME format and readable by any OpenPGP-compliant encryption software, such as GnuPG. You can open the file using a text editor like Notepad, but it will mostly be random-seeming numbers and letters.
The signature can be verified against my GPG public key, id A8CD83A7, fingerprint D4B3 F8EE B115 ADEC A04E D9D9 45A7 167F A8CD 83A7, which can be found on the pgp.mit.edu keyserver.
I use and recommend the Enigmail extension with Mozilla Thunderbird.
If that was mostly gobbledygook to you, then the Wikipedia page on digital signatures is a good starting point for learning more.