If you want to send emails from a form it would be a good idea to learn about server-side scripts.
I would start by finding out what software you have installed on your current server. Eg: is it a Linux server with Apache, MySQL, PHP, Perl etc... or is it a Windows server with ASP.Net etc... You can find this out from your host provider.
For security reasons I would not recommend writing your own script for sending emails until you know what you are doing. There are freely available scripts that do this fine.
Tectite FormMail is one example if you have PHP
Then you need to be able to upload your required scripts to the web server. Usually FTP access to the server is given for this task. And use the appropriate HTML to communicate with the script.
Alternatively:
There are hosted form options that may be worth checking out so you don't have to learn as much about server scripts. (this means some other web server does the email part all you manage is the HTML)
Here is a list of a few hosted options
Edit: Here is a better list of hosted options