You can use a dynamic IP to host your site, but you will need another company to point your domain to your dynamic IP address. The most famous one is
http://www.dyndns.com/
It is free if you use one of their domains. If you want to keep your domain, there is a yearly charge. You will need to choose the "custom DNS" option. You might find other companies that will do this also.
You can't choose to use a static or a dynamic IP address. Unless you have a very nice ISP. You have to use what your ISP gave you, most likely a dynamic IP address. But, if your ISP gave you a static IP address, then you won't need the other company.
Now, my pc's have static IP addresses, because I set them up that way. But these are local network IP addresses for my LAN. My router knows which computer has which IP addresses, but my router's IP address changes every so often, not sure how often. If I was connected directly to the modem, then I would need to change to a dynamic IP address on my PC to connect to the internet. That is probably why you couldn't connect to the internet when you tried to use a static IP address before.
Ok, you are behind a router, but your router most likely has a dynamic IP address.