I'm going to gloss over some details to keep this simple, but a script is like any other file on your server. You upload the script file to the server with your FTP program. Then when the server receives an HTTP request for the script, it notices that it needs to be executed like a program instead of simply sent to the user "as-is" like an HTML file would be.
Most of the scripts that you'll find these days are written in a programming language called "PHP". You should consult the documentation for your hosting service/server to see if they support PHP (which is almost a certainty unless you're on a free host), and if they have any special requirements or restrictions. On the whole, you should be able to use any conventional formmail script. Formmail scripts are generally designed to be used with feedback or contact forms on websites. They just gobble up the <form> data submitted by the user and send it to a designated Email address. You'll have to edit the script with a text file editor so that it knows your Email address and, possibly, some information about your website. Then just upload the script to the server and set the "action" attribute of your <form> tag to the URL for the script file. Good luck!