Just a personal website, not anything business related.
My computer is behind a router, that means multiple devices (like other laptops, ipads, phones) share the same IP. Is that still possible?
Can anyone point me to a tutorial?
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Yes it is possible, but you'll need to address a couple of keys issues such as ensuring the computer you're doing it on, and your modem/router, has a constant power supply - one that is separate from you're home supply. If your house has a blackout, then you run the risk of your web-site going offline for an undetermined period.
Likewise if you need to do some remedial electrical work to your house which requires the power being off for a period of time, then you'll have the same issue.
http://dyn.com/dns/ is one company the allows to you set up a dynamic DNS for your home PC so that you can have a web server.
Once you have a domain and an IP address set up for your server, then as far as I can see it's just a matter of setting up a web server on your PC. If you use Apache, then setting up the LAMP stack on an Ubuntu machine is very easy to do.
I've never set my own home web server so there may be other steps (that is a viewable to the outside world). I've looked into setting up a home web server before, but when weighing up the pros and cons v a dedicated server with a web host, I've always gone the latter option. There is a lot that can go wrong with a home web server and most of it is going to be unforeseen so dealing with it can be a pain.