The 9 Volt battery is an odd addition to the household battery family. AAA, AA, C, and D batteries all look like scaled versions of each other, cylindrical dry cells with a positive terminal at one end and a negative terminal at the opposite end. Furthermore, they are all 1.5 Volt batteries (1.2 Volts for rechargeables). Their size variations only indicate the amount of energy they have stored in them.
The 9 Volt on the other hand is rectangular with both terminals side by side at one end, designed to snap securely into place. The radical change in design is explained by unwrapping the outer cover: The 9 Volt is actually made up of 6 small batteries that look a bit like miniature AAA cells, arranged in a six pack formation and connected in series.
Standard alkaline dry cell battery chemistry provides 1.5 Volts per cell1. The other common batteries are made up of one cell, that is, one anode and one cathode separated by a chemical that encourages electrons to travel between them. This encouragement creates a potential voltage difference of 1.5 Volts between the anode (the positive terminal) and the cathode (the negative terminal) as some of the electrons are moved from the anode to the cathode. Hooking the battery up to something, for example a flashlight bulb, allows the electrons to travel from the cathode back to the anode, doing work in the process. As a side note, for historical reasons we define the direction of the electric current as opposite to the direction of electron flow. So although the electrons travel from the cathode to the anode, we say the current flows from the anode to the cathode.
Since very few electric devices operate on 1.5 Volts, most use more than one dry cell battery. A remote control for a television for example typically uses two AA batteries connected in series (that is, with the anode of one battery connected to the cathode of the next) to produce the 3 Volts necessary to operate the remote control. While this concept could be extended indefinitely to get higher voltages using more batteries, weight, space, and convenience become an issue when more than four cells are needed.
So to get 9 Volts out of one battery, six small 1.5 Volt cells are connected in series and wrapped up into one package. 1.5V x 6 = 9V. 9 Volt batteries are used in electronics that require higher voltage but not much current. Typical applications are in R/C toy controllers, metal detectors, and walkie-talkies. These devices all work by creating electromagnetic fields, which is more of a function of voltage than current2.
Things you can do with a 9 Volt battery:
Motorized toys that require 2 AA batteries will, with a little tape and aluminum foil, accept a 9 Volt battery and run much, much faster. Be careful to get the polarity right, don't short the terminals to each other with the foil, and be aware that you may very well burn out the toy's motors because you are applying three times as much voltage as it was designed to take. And never do this to a non-motorized device, all you'll do is damage it.
Touching a 9 Volt battery to your tongue will tell you if it is still good; 9 Volts is enough to produce a gentle tingle on your tongue if the battery has some charge in it. Be careful if you have a lot of dental work.
Notice that the snap-on terminals of the 9 Volt battery are identical to the snap-on connectors that connect it to the electronics. Two 9 Volt batteries will snap into each other, creating a powerful short circuit between the two batteries. This will produce a lot of heat, so don't leave them connected for very long. Also, the batteries will be damaged and may leak if you do this.
Connect a Ni-Chrome wire to the terminals (dangerous, see the writeup by 3Suns).
Start experimenting with hobby electronics. A simple and cheap but inefficient power supply can be made with a 9 Volt battery and a Zener diode for experimenting with parts you can get from Radio Shack. Start with a breadboard, and later learn how to etch your own circuit board.
1. Technically, a single 1.5 Volt "battery" is called a dry cell. Multiple cells in series are called a battery. So a 12 Volt car battery (made of six 2 Volt wet cells) or a 9 Volt battery (made of six 1.5 Volt dry cells) are true batteries, while a AA is just called a dry cell. Common usage however has extended the definition of the word battery to any self-contained chemical package that creates voltage.
2. Note that a television remote control uses an infrared emitting diode, not radio waves, to send signals. This allows them to use the lower voltage AA or AAA batteries, usually two in series for a total of 3 Volts.
Wiccanpiper says: "I strap ten of 'em in series and use that to power my antique tube-type battery radios. Most only draw 20-30 mA, so the batteries last quite a while."
Wertperch tells me that 9 volt batteries are commonly called "PP3" in England, after the "power pack" series designation given to them by the Ever Ready (not to be confused with Everready) brand of batteries.