What is RPM - Tachometer

The RPM meter, or Tachometer, on your dashboard is used to show the driver how many Revolutions Per Minute (it shows you the speed of your Engine)  your engine is rotating at.  This is to show the driver when they are approaching the mechanical limits, redline, of the engine.

If a driver in a manual transmission car does not shift before hitting the Redline of the engine they could damage their engine.  Automatic transmissions typically protect the car by shifting before the redline automatically.  Also in modern computer controlled cars there is a fuel cutoff at Redline where the computer will stop feeding fuel to the engine before it would be damaged so the driver cannot accidentally exceed the redline.

Now that being said there is still a way to damage the engine if a driver of a manual transmission car is not watching their tachometer.  If a driver were driving close to redline and downshifted to a lower gear the engine RPMS would increase very quickly and exceed the redline of the car and the fuel cutoff would not be able to stop this.  The engine could be damaged by the driver this way.

Depends.

If you are driving stick, you are going to end up using it a lot more than you would if you had an automatic.  The reason being that, the RPM meter, or tachometer, in a way measures how hard your engine is working.  You would usually use it to pick when to shift (probably around 2500-3000 for daily driving).  As far as automatics go, its really no use except for if you like to watch those kinds of things and get an idea of your input as a driver.  The other thing I can think of is, say you start your car in the cold one morning, and the car stalls (shuts off), you might have a better chance of saying it was due to idleing problems than to assume it was something else.  If you watch your tach in that situation, it might be more clear to you (chances are you would need new spark plugs then).