Packet radio is a means of doing computer networking via ham radio. Any licensed ham can do packet provided they have the proper equipment. On the 2-meter band (one of the most common bands around), packet speeds can reach up to 9600 baud though 1200 and 2400 are the most common. This is because of the ready availability and cheap price of equipment.
Of course, this is not the upper end of packet speeds. 19.2k and 56k connections can be established on the higher frequencies though equipment gets more expensive and more stuff will need to be built by hand. The upper limits seem to be over 1 megabit/sec via microwave links. This is where you'll have to build most everything yourself. Ongoing research in Europe has pushed the limits closer to 30 mpbs. I'd just like to pause for a second and say that this is all travelling over the air! Amazing!
So, what's needed to do packet radio? I'll assume that you're sticking to the 2-meter band since that's what I've done most of my research on. You'll need a radio and an antenna, of course. A handheld radio will do just fine for short range communications. You'll also need a computer with the proper software and a serial port. Finally, you'll need a TNC (terminal node controller). This is the device that sits in between the computer and radio, converting network traffic into radio waves. 2400 baud TNCs should be cheap and easy to come by these days, and you should already have a radio (otherwise you wouldn't be a ham!) and a computer (or you wouldn't be on everything!).
What's the right software? Well, Linux has support for packet radio built right into the kernel and there's active development of all the userland tools needed. With Linux running on your computer, you can run TCP/IP over your packet radio link. These are the same protocols that power the rest of the Internet so in effect, you can talk to wired machines all over the world. Further, there's an active community of radio-based BBSs, mail transfer, and so forth. Other protocols are used (ROSE, NET/ROM) but like in the wired world, TCP/IP appears to have won.
So you should be able to set up a packet radio station for very cheap, get the software set up (there's a howto if you're interested in running it off a Linux machine), and start seeing what's going on in the airwaves all around us. Here in Atlanta, there's a group of people running the East Atlanta LAN which is a network of machines running over packet. Some of these machines will also provide access to the rest of the Internet via a DSL connection or other fast bandwidth.
One final note is that packet radio was assigned the entire 44 class A IP address block. This block is then sectioned out into class B networks by geographic region. For example, Georgia (in the USA) is assigned the 44.36 network. This is then split up by a local coordinator and each permanent packet station gets a real IP address.