IP addresses come in two different formats now, either groups of 4 or 6, numbers separated by "." periods; ranging from 0 - 255. IP addresses are used to connect devices to the Internet. Knowing what your IP address is will allow you to utilize different programs like GoToMyPC, PCAnywhere and UltraVNC for desktop sharing. You can also use your IP address to host Internet games.
For the most part if you use DSL or Cable to connect to the Internet you will have a some what static (won't change) IP address. I say "some what static" because I have found that my Internet provider changes my IP address every so often and if you loose power to your DSL or Cable modem it might obtain a new IP address then also. If you are still using a dial up connection for Internet access your IP address will change each time you connect to the Internet.
Because IP addresses are provided to Internet providers and Hosting companies in groups or blocks, we are able to define more closely the Geographic's associated to IP Addresses. Web sites, ISP's and Government agencies are tracking not just the specific IP address associated to your visit but also where you are located in the world.
If you are trying to connect to a computer behind a router you will probably need to set up a port forwarding rule to be able to connect to that specific computer. Most computers come with firewalls these days, so you also want to make sure your firewall will allow the communication too.