Why Pistons?
Simply, pistons do not fizzle out like redstone torches. It is therefore possible to create circuits with only pistons, redstone wire, and repeaters, as these can run at a third of the speed as traditional circuits.
The Principle
Power is transmitted in several ways that are useful to pistons. The first thing to note is that there are two types of solid block; transparent and solid. Transparent blocks are things such as glass or air, and solid blocks (the most common) are things such as dirt and stone. If a solid block is on top of a redstone torch, any wire connected to the block will be powered. If, however, the block is transparent, the torch will not power the wires.
When a repeater is directed at a solid block, it will pass power into that block in the same way redstone torches do. Power will not be transmitted by transparent blocks.
There is another related detail:
These two above will work. However,
will not work
Simple logic gates
If you don't understand these, look at Redstone circuits.
NOT Gate
FYI the piston is sticky. However, if you want to invert things, the easiest way is to do it is to run redstone wire into a block with a redstone torch on the other side of it. 2x2x1
Red = Input Green = Mechanism Blue = Output.
OR Gate
The piston is sticky. If any of the inputs are on, the output will be on. 3x2x1
Red = Input Green = Mechanism Blue = Output.
AND Gate
Note that the piston is sticky. When both inputs are on, the output is on, too. 2x2x1
Red = Input Green = Mechanism Blue = Output.
IMPLIES Gate
A device which represents material implication. Returns false only if the implication A → B is false. That is, if the conditional A is true, but the consequent B is false. It is often read "if A then B." It is the logical equivalent of "B or NOT A". 4x2x2
Red = Input Green = Mechanism Blue = Output
XOR Gate
A device which activates when only one input is on. Pronounced "exor", and is a shortening of "exclusive or". Adding a NOT gate to the end will produce an XNOR gate, which activates when the inputs are equal to each other. A useful attribute is that a XOR or XNOR gate will always change its output when one of its inputs changes, allowing for 2 switches to be combined to open or close a door, or activate another device.
The Piston XOR gate is much more efficient compared to an XOR gate without pistons.
3x2x1
Red = Input Green = Mechanism Blue = Output
More Complex Machines
These are very useful and very compact. In some cases, far smaller than standard redstone circuits.
Clocks
Here is a very simple gate. Each line leading out of a repeater is an output. It can also be switched on and off.
Red = Input. Green = Mechanism. Blue = Output.
RS NOR latch
This RS NOR Latch (aka. memory cell)is easy to make and has two outputs just like a normal RS NOR Latch, but on the same side. The outputs can also be on the same side which can make things easier. 4x2x2
Red = Input. Green = Mechanism. Blue = Output.
Pulse limiter
Here is simple piston based pulse limiter size of 3x2x2. Works great and can easily be used. 2x2x1
Red = Input. Green = Mechanism. Blue = Output.
T-Flip Flops
This is a very tiny design of a Piston T-Flip Flop that works good. It's made small to fit in to tiny places, but I prefer the design under this one which works better and is smaller. Dimensions are 4x3x2.
Red = Input. Green = Mechanism. Blue = Output.
This is a small T-Flip Flop that is absolutely great and is really fast. It's one of the smallest One of the fastest there is.
Red = Input. Green = Mechanism. Blue = Output.
Rings
A band is a ring of blocks attached to pistons at the corners so it can rotate. The blocks are usually a combination of solid and non-solid blocks. The pistons are often connected to a clock so that they will rotate the band. By using a band, you can create things like item sorting machines and other complex mechanisms.
Bands
When you add several rings together in a row, you create a band. A band is useful for even more complex things, as it can be used in a similar manner to punched tape. Examples include music machines, combination locks, and memory.