It can be done, but you need to know what you're doing if your site will see any kind of real traffic.
I host multiple sites from my ATT cable broadband connection, which has a "dynamic" ip address. In reality my IP has only changed once, and that was when ATT was doing a major expansion of their network in my area. If you do it this way, you need to have your domain names on a DNS server that synchs very quickly in case your IP address changes. Also, what I'm doing is strictly forbidden in the ATT terms of use, so proceed at your own discretion. I don't believe any broadband ISP actually allows you to run servers unless you pay for a higher business class connection. I'm getting to the point where a burstable shared T1 would actually be cost effective as I get paid to host/develop sites. That is really the best way to run a server... but I'm proof that it can be done using a standard residential broadband connection. As Elizabeth and dalecosp noted, you need to research your ISPs first to make sure it will actually be possible. I have quite a bit of experience with 3 different ISPs, MSN, Covad, and of course ATTBI. MSN dsl should be avoided at all costs. You will not be able to run a server on their service... no way, no how (I know because I actually worked for them for a while). Obviously ATTBI will work, but there's a lot of drawbacks... the main one being you'll get shut off if they find you out. Covad on the other hand is in my opinion the best way to go. They have service plans that are reasonably priced which actually allows the use of servers in their terms and conditions.
As far as security is concerned, that's a whole other can of worms. If you're really serious about security, only use Unix (solaris, irix, etc), BSD, or Linux. Windows can be secured effectively, but you really have to be on your toes. If you're going to use winblows (2000 server is really the only acceptable server os from M$) make sure you know about, and know how to correct all of the various security flaws and exploits that exist on windows systems. You'll need to do your homework in this area though... I suggest http://google.com. 😃
Good luck.