Hi I run my own website and my router accepts incomming connections and I am on AOL uk but does anyone know if it against there TOS to host a website on my computer and link my domain to the ip? Thanks

AOL and most other ISP DON'T allow you to run a webserver on your own box on a domestic connection.
For starters you'll need a static ip address for this to work correctly, which you'll need to spend more money on. Basically what you really need is a business orintated ISP that will allow this kind of connection.

Cheers my ip is normally static as when I turn my PC off the router is still connected so I've seen it the same for about 3 days before then rebooted the router because I watch football on the net and it slow the router and internet down loads if you don't restart after.

Thanks anyway

Your public IP probably isn't static, you would know if it was as you would be paying more of it, plus I'm sure AOL doesn't do statics for domestic use.
Your computer's IP may well be static in the 192.168.x.x range, but this isn't a public address.
Your public address is in the range 172.215.x.x.

If you really want to run your site have you looked at http://www.no-ip.com/ may help you get around the dynamic public address.

thanks I know my private ip is static as I reserve 192.168.0.2 for my laptop in the router config page but my public 172.200.x.x normally stays the same for a while even if I turn my PC off it only changes if I disconnect or reboot my router

As your router deals with the connection to your ISP this is hardly surprising.
Your public address may seem to be static, but I can assure it isn't. They do change periodically as you've said whenever you reboot your router and usually every 7 days or so.

Thanks ill stick with the hosting as I can't be bothered to change it eevery 7 days and it is slower because of my limited 512kbp/s upload speed.

Thanks

Several points here. With most broadband connections the external address (provided by your carrier) is not static. You usually have to pay more for a static IP. While it may not change (mine hasn't for a couple of years), it is possible that it will.

AOL definitely doesn't permit running a server over their broadband connections like most ISPs. This uses up bandwidth.

I would definitely not recommend running a commercial site yourself. Considering security, maintenance, connections etc, it is almost always more cost effective to use a host service.