Generally it is a small concealable auto loading pistol of limited power. It takes its name from the .25 ACP cartridge that it fires. It is called .25 caliber (notice the decimal point?) since the...
Laws vary from state to state. In MOST states, there is no "registration" of firearms as such. If you live in a state that DOES require registration, then yes, that would be illegal.
Very common. The .22 pistol is a very popular target and training firearm, and is a sizable part of the handgun market. There are about 10 in my collection.