Within an isolated network, you can assign IP addresses at random as long as each one is unique. However, connecting a private network to the Internet requires using registered IP addresses (called Internet addresses) to avoid duplicates.

The four numbers in an IP address are used in different ways to identify a particular network and a host on that network. Four regional Internet registries -- ARIN, RIPE NCC, LACNIC and APNIC -- assign Internet addresses from the following three classes.

The number of unassigned Internet addresses is running out, so a new classless scheme called CIDR is gradually replacing the system based on classes A, B, and C and is tied to adoption of IPv6.

Also see Understanding IP Addressing in the Did You Know . . .? section of Webopedia.

TCP/IP and IPX routing tutorial 
This tutorial supplies information on setting up a relatively simple WAN-connected internetwork or Internet-connected LAN. Includes explanations of IP addresses, classes, netmasks, subnetting, routing, several example networks, and a basic explanation of IPX routing

LearnToSubnet.Com
An educational course on addressing TCP/IP Networks that includes IP Addresses and Subnetting. Topics include: Binary Math, IP Addressing (IP Address), Subnet Mask, and Custom Subnet Mask.

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