Yes, they are all die cut machines.
The Sizzix is a manual machine. You buy individual dies that make 1-4 shapes. There are a couple of variations. "Big Red," the dies are about 1/2" thick and sell for $5-$15. It has a big lever that you move up and down to cut things. You can get an adapter to cut Sizzlets with it. Because of the steel dies you can cut a wide variety of materials (paper, chipboard, vinyl, fabric, fun foam). However, since they came out with the Big Kick, some dies won't work in the Sizzix. I'm not sure how many places still sell this one and just sell the Big Kick instead.
The Sidekick is smaller, it uses "Sizzlets" which comes in packs of 4 for $20, or alphabets that are very expensive. The sizzlets aren't very thick. You feed it into the machine and turn a crank to cut things.
The new machine is the Big Kick, which uses the thicker dies, but you feed it and turn the crank. You can use any of the dies with it, including the Cuttlebug dies.
The Cuttlebug is also a manual machine. I've never used one but I think your feed and turn a crank too. And I think you can use the Sizzix dies with it, although I think some people are afraid to use the big dies because part of it can break. The company that makes the Cuttlebug used to co-own the Sizzix but when they sold out they came out with the Cuttlebug. They also make the Cricut.
The Cricut is an electronic cutter. You buy a cartridge that has a wide variety of shapes or letters. You type what you want to cut in the keypad and then it cuts it out for you. The cartridges retail for $90, but there are lots of sales to get them between $45-$60.
