You can do that using a Server-side language (SSL) such as PHP or ASP and a
database, such as MySQL.
In an extremely basic description:
You would have one page with your list in it.
Elsewhere in the page would be a Server-side include (SSI).
When you select a specific letter and click a submit button, this
information is transmitted to the server and the corresponding Video Games
are selected from the database, then displayed where the SSI is located when
the page reloads.
You can get some information here to start:
Judging by many, many posts here recommending a certain book as well as
having read the author's solutions in many threads here, I don't think you
can go wrong picking up David Power's "The Essential Guide to Dreamweaver
CS3 with CSS, Ajax and PHP"
It's worth checking out at: