[politician]

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.

[belorn]

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.