Basically:
The internet is kinda like a giant phone book, with each computer connected to it having it's own "phone number" AKA IP address. There are servers that correlate your name request (http://www.colmbusracing.com) to it's IP address (65.118.247.22). So, clearly, you gotta have a lot of addresses and most computers and devices on the internet cannot share an address - they must be unique just like a phone book.
The internet is widely estimated to run out of IPv4 (I.P. version four) addresses some time between 2010 and 2012 with the end-game being March 2012. IPv4 is 32 bit which allows 4,294,967,296 (2 to the 32nd power) possible addresses, IPv6 is 128 bit allowing 2 to the 128th power, or a giant flying fuckload more than IPv4.
Most business and home networks as you know them will not change, and you won't need any updates or other crazy hoobajoob to the average computer - at least not for a while. IPv6 will impact the big companies that bring you the internet such as ATT, Time Warner, etc. first and most because these are the guys taking up internet address space. You see, with your home DSL connection even though you have eight computers, you're only taking up one internet IP address - that's what the router thing does for you. Eventually you'll get an IPv6 address, but you probably won't notice or be impacted at all because new DSL/cable modems will be shipping with IPv6 support and they'll gradually phase it in.
That about cover it?