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:
openssl ecparam -genkey -name secp256k1 -noout -out myprivatekey.pem
To create the corresponding public key, do this:
openssl ec -in myprivatekey.pem -pubout -out mypubkey.pem
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
-pubout if you want the private key, leave it if you want the public key. And you can use
-outform DER to get DER format. You can use
-text to get hexadecimal.
Do not give any form of your private key to anyone else.