This is often called CGI or Common Gateway Interface. The webserver is actually a program running somewhere on a host computer. This normally doesn't do anything but serve web pages. It will pass on the form data to another program running on the same host. That program may communicate with a database, a third program on the host. Depending on the outcome there may be feedback to the origininating web browser.
Form output is often handled by some kind of script. It is possible to write dedicated software for this, but scripts are generally easier to write and maintain. Even with very little programming knowledge, you can write some nice scripts. Whereas dedicated software requires a lot more experience. Scripts are also very portable, a script will often run unaltered on a windows as well as a mac or unix based machine.
The most popular scripting language is PERL, used originally for processing large amounts of text on unix computers. Since then ported to other operating systems. There are a lot of free scripts available on the web, as well as tutorials and references.
Lately there are other languages coming up, PHP looks very promising. In particular in combination with database software. If you run a windows server you can use ASP or Coldfusion. Tcl/tk is also quite popular, mainly on unix machines. I use PERL, as this is the most widely available.
You may have gathered by now, creating forms is only half the story. You will need to acquire some programming knowledge. I would advise you to start out with a scripting language, since these are easiest to learn, and most portable. You can however use any programming language you are comfortable with, just as long as it supports CGI standards.
I would advise you to use PERL, since that is most widely used and available. Just use any search engine and type in "perl tutorial", and you will find hundreds of sites. A good second is PHP, relatively young, but very promising.
If you are serious about web programming, I would advise you to get familiar with databases. In particular SQL databases, since these are most portable. Also some good ones are available for free, on a variety of platforms. Look for MySQL or mSQL. To run scripts locally, you will also need a web server.
Most times cgi, perl, or database software will not be available on a standard ISP account, you will probably need some dedicated web space. Some free web space providers will offer cgi support, although most of them will not, or only very limited. Databases are even harder to get.
Related links are in the webmaster resources section.