Button

The button is a non-solid utility block that can be interacted with to provide temporary redstone power. These buttons can be crafted from stone or even wood. They are naturally generated in a stronghold right next to iron doors.

A stone button placed next to a door (via screenshot)

How to get a button?

You can either break naturally generated buttons in a stronghold or you can craft them. The wooden ones are easily broken with any sort of ax but a pickaxe is required to break a stone button. They are also naturally generated in village huts in the desert biome. Crafting a button is easy, you will just need a wooden plank of any wood type, or a stone block, or Blackstone.

Making a button with an Oak plank (via screenshot)

What is the use of a button?

A button can be used as a monostable power source. They provide a temporary redstone signal that cuts off after a short time. They can be placed on any surface and on the side, top, and bottom of any fully opaque block.

They are usually in an inactive state but can be temporarily activated by players by using it (right-clicking it). Upon activation, the button will remain active for 1.5 seconds and provide redstone power for that duration. After 1.5 seconds it will return to its inactive state. While the button is active, it will:

power any adjacent redstone dust up to power level 15, including any dust beneath the button.
Power any redstone comparator or repeaters that are facing away from the button.
It will strongly power the block it is attached to.
Activate any adjacent mechanism components such as doors, pistons, etc.

Conclusion

Buttons are a great way for redstone circuits that are only required to be active for a short amount of time such as automatic doors or piston doors. They aren't widely used since the duration of their activated state is really short.