It depends on what type of game you would like to host as well.
Different games have a server that you can run on your computer that
everyone else can jump on and play. For example, if you are playing
Counter-Strike, you can install the Steam Server (From your Steam
menu, you should have an option to install a "Dedicated server". Once
you click this, you can adjust the settings to whatever pleases you
and it will automatically be on the game servers list. Other games do
it different ways. If you could specifiy which game you are talking
about, we can provide a more concise answer.
Also, when running game servers, you need to make sure that you have
the capabilities to provide a good enough service. What I mean by
this, is when you are connected to other people from around a world on
a single server, that server needs to be able to handle all that
traffic. You need to have a powerful enough computer to be able to do
this. You also need to be able to provide a good connection to
everyone who comes to play. For this, you need a good internet
connection. Depending on the game, a DSL connection may be sufficient,
but if you go back to our assumption of running a big game like
Counter-Strike, you would probably need a T1 connection.
Hope this helps.