The new Observer block added in 1.11 was introduced to keep block update detectors in the game while fixing the bug that made BUD circuits possible. But surprisingly, the least popular use of this block is detecting block updates.

The Observer block is very simple, all it does is detecting a block update on the block it is facing, then sending a 1 tick pulse at strength of 1 with no delay to the opposite direction.
The no-delay part is what made these blocks such a powerful redstone component, but many of it's uses are also based on the other properties, so everything is important.

I listed here the uses the Observer block has in redstone, please let me know about any uses I didn't list here so I can add them.

Observer block uses:

• Sending redstone signal through a 1x1 conductor. (instead of redstone dust that makes a 1x2 conductor since you must have a block under the redstone dust)
  
• Sending redstone signals through one air block (great for when you need some block empty but also need to pass a redstone signal through it, that solves a lot of compacting problems)
  
• Sending instant redstone signals across long distances to any direction. (you can send redstone signals horizontally and vertically with no delay at all, using a row/pillar of observer blocks alternating with air blocks)
  
• Generating a 1-tick pulse.
  
• Generating a weak redstone pulse (power of 1, useful for powering redstone dust without powering adjacent redstone dust)
  
• Multiplying pulses (outputting 2-3 pulses when receiving only 1)
  
• Bidirectional wires (alternate observer blocks facing opposite directions in a row/pillar, then the arrow in the texture of power the block with the arrow of it's texture is pointing towards the direction you want to send a signal to. Note that the edge observer blocks must be facing outside so that you will be able to take the signal out at the very edge conveniently. This is incredibly useful in compacting redstone devices and solves the problem of bidirectional repeaters)
  
• Creating a source of constant power that powers only one specific block (there is a bug that allows you to make the observer block to keep powering the block it is facing until you break and replace it, simply move the block with a piston right when it detects a block update)
  
• Solving many signal-transmission problems when compacting devices, also enabling the creation of many machines that weren't possible before.
  
• Compacting devices, making them extremely small compared to what they were before.
  
• Making devices work faster (especially big devices like computers)
  
• Transferring items instantly through droppers
  
• Building the most compact and fastest engine for flying machines (2x2x2) with an option of making it bidirectional just by replacing one block. (more info and guide here)
  
• Creating very fast and compact elevators (~30 blocks/second, 1x3 blocks/layer) with an option to stop at any floor in the middle.
  
• Making a very compact (1x2x3) and conveniently adjustable clock.
  
• Creating a high resolution screen where every pixel is just 1 block, without using command blocks!
  
• Compacting efficient auto-farms, and other uses of detecting block updates... (I even forgot to add that one to the list at first lol)