You need a form handling script, which takes the information the form user inputs and turns it into an email which is then sent on to the address you specify in the handler script.
Most people use either a CGI (Common Gateway Interface) or PHP (Hypertext Preprocessor) script.
Your choice could be governed by what the web host supports where the site is currently hosted. If you are allowed to run your own CGI scripts and you have a CGI bin you can obviously choose to use a CGI script. Where the web host supports PHP you can opt to use that instead.
PHP scripts are usually easier to get your head around than CGI when you are first starting out. Most PHP scripts need no file permissions assigned to them, while pretty much all CGI scripts need permissions assigned to allow them to run using CHMOD on Apache servers, or by using the built in permissions tools in your FTP client.
Here are some links for you to try for PHP form handlers:
And for CGI scripts: click here
I'd plump for PHP if your web host supports it. Instructions for use are on the various script sites and included when you download them.