Let's say I want to have a second domain name point to the same web site as the original...does that require a new IP address? If not, is there any advantage to doing so?

Thanks!

Hi!
If you just want to point it to another domain ... there is no issue involved... It's called domain parking. Just create a DNS record for point to this IP and add the mail settings...

If you want to host domains with the same IP... that too is possible... in apache it is called named based hosting.
I'm assuming that you are using apache as your webserver...
You can get more information on name based virtual hosting here

http://httpd.apache.org/docs-2.0/vhosts/name-based.html

Have a great day :)
Regards
amar