If you don't have any other PC at home, you don't need a router. You can build a firewall on your machine and will have protection (a firewall with NAT on the router would not be useless of course but I don't think it is a very big issue if it is a home server not dealing with financial or users info).
Now, regarding the ISP, some ISP do not allow many computers connected but this is not your case as you will use only machine (with 1 IP).
The other point regarding ISP is that many of them block the ports usually used by servers (80, 25, 110 for example) and your server (with or without router) won't be reachable on regular ports (you will need to include the port like this
http://www.site.com:8080 to reach it. There are a few workarounds (nothing very nice) to that like an external port forwarding service for example.
To test if your ISP blocks the port you need without asking them (I saw your comment ;)), you can put something to listen on port 80 (you didn't say your OS but for linux
nc -l 80 should do the trick or something similar) and from an external IP, issue
telnet your.ip 80 and see if it answers.
Anyways, this is not a very good way to host something more than personal data as if your machine breaks (or is stuck with an update) or your Internet goes down (or many other things), your server will be down. Have a look at your ISP speed limits too as usualy home internet is not adequate for servers. The download speed is ok but the upload maximum speed is normally a lot lower.