I want to know because I would like to generate a vanity address at Vanity Pool.
Install OpenSSL. When the executable in your path, enter this command to generate a private key:
To create the corresponding public key, do this:
This will give you both keys in PEM format. I'm not sure what format the web page wants, but it shouldn't be difficult to convert. You can use variants of the last command to output other formats. Remove
Do not give any form of your private key to anyone else.