If you set up a button, a piston and a redstone dust in this composition...
The piston will be activated while the button is on (and will retract when off).
However, I can't find the rationale why this gadget works. I guess this is some kind of BUD mechanism, but still don't understand what route the piston is powered through.
As far as I know, if I remove the dust below the button, it'll no longer trigger the piston.
Nor when the dust is powered otherly.
This is a block update phenomenon. The button is providing diagonal (BUD) power to the piston, and the redstone dust is providing updates to the piston, which causes it to extend. Normally, block updates are only triggered on the 5 sides of a piston. However, redstone updates are special, and can provide updates to a piston 2 blocks away.
The dust doesn't provide any power to the piston (which is why the piston doesn't extend with just the dust in this configuration), however, it does provide a necessary update to the piston (which is why the piston doesn't extend without it).
This is called quasi-connectivity, it is a property that is shared by pistons, droppers and dispensers. Basically, what it means is that the redstone appliances that are effected by this phenomenon can be activated by having the block above them powered, this includes air blocks. This may seem like a bug, but the team at Mojang have stated that it is a feature that works as intended. It sometimes causes trouble in circuitry, however it is the premise that many piston doors work on. Without this feature, flush piston doors would be incredibly complex and difficult to build.
You can find more information on the subject, as well as tutorials, on the Minecraft Wiki.
