The messages that your computer sends to the internet include an IP address which the internet uses to send responses back to your computer. The problem is that these addresses are unstable -- your ISP reserves the right to change it, basically.

If you want to host a server from home, you have a few choices:

a) Obtain a static IP address from your ISP.

b) Obtain a domain name from a dynamic DNS provider and map that to your current IP address. Update the mapping as necessary.

c) Just use your existing IP address -- this is how P2P or Skype works, for example.

There is an alternative, one can create a IPv6 only service, as tunnels with static IP are commonly free.

Of course there is the downside that no one except those very few people that has ipv6 will be able to use that service, but it is free and it can be a very useful way to access a computer behind nat.

It really depends on the ISP. A static IP generally has to be requested, and at a cost. Additionally, some ISP's like to be restrictive blocking inbound low ports such as 25, 80, 443 - so hosting a web server off your home connection might be a problem depending on the ISP.

It's specifically against residential Comcast ToS, for example, to host a publicly-accessible website.

If you have a dynamic IP, Dyn offers DNS services[0] that let you tie a domain to it through them. I believe it requires you to install something on your machine that monitors your IP and reports it back.

Otherwise, you just use your IP and make sure to edit the DNS yourself when it changes.

[0]: http://dyn.com/dns/

Many routers have dynamic DNS support built in now. For example, my Asus RT-N66U supports several DDNS services including Dyn and a free one that Asus provides for its router customers at {yourcustomsubdomain}.asuscomm.com.

No big surprise here, since the router is really just a nice little Linux box.

Ah, I have an Asus RT-N16. DD-WRT is truly awesome!

That would be something like: http://dyn.com/support/clients/linux/ddclient/

By your ISP. Some plans come with a static IP by default, but many providers make you pay extra (or go onto a business plan if you tell them you want to run servers)