Everything on the Internet must have an IP address to work. "IP" stands for Internet Protocol. The IP address is a unique identifier, rather like a street address for a home, that allows web traffic to be properly direct. Hence, just as a street address on an envelope tells the mail delivery person where to take the envelope, the IP address of an Internet communication directs the communication to its proper destination.
Thus, when an individual is browsing the Web, both the individual and the site he is visiting have a unique IP address, allowing communication to go back and forth without confusion.
IP addresses can be permanent or temporary. Websites often have a permanent IP address which always leads to that specific site. Individual web surfers may have a permanent IP address, but more often have a "dynamic" IP address assigned automatically when they connect to the Internet. If they disconnect, the dynamic address will be reassigned to another user.
It can get more confusing. Websites often share IP addresses with other websites, using additional information to direct traffic to the correct place. (Think of an apartment building whose residents share the same street address but get their mail delivered properly.) Although the sharing of IP addresses poses no technical problems, SEO experts usually suggest a unique, individual IP address for each site. This reduces the probability of being mistaken for a spam site should a shared-IP site engage in practices frowned upon by search engines.
Similarly, at a site where multiple individuals connect to the Internet, there may be just one IP address for the location; each individual, though, will have an internal IP address which differs from the single external address. The router or other connecting device keeps track of the internal IP addresses, routing each communication to the correct user.
IP addresses take the form of four number groups separated by periods. Due to the ever-increasing number of websites, Internet users, and other connected devices, the number of IP addresses in the current scheme will eventually run out. A new scheme with more digits and groups will eventually solve the problem of running out of IP addresses.