An IP address is an address for a computer on the Internet. The usual example used is of a web server that can be accessed via a URL that is translated behind the scenes into an IP address, but IP addresses can be used to find any computer on the Internet.
When a normal home broadband user accesses the Internet they send their transmission through their Internet Service Provider (ISP) who have a collection of IP addresses they use for their users. ISPs tend to get blocks of perhaps several thousand IP addresses that they will use as a pool for their users. When a user logs on they are given an IP address and when they log off this address is sent back to the pool for other users to use. The actual systems in use here are a little bit more complex than this, but this is the essential idea.