Why do rockets have multiple stages? Are boosters an essential part of every launch vehicle?

Rockets have multiple stages because the effectiveness of a rocket is inversely proportional to its mass and using stages allows us to reduce the mass of the rockets as it operates.

Tsiolkovsky's rocket equation tells us:

In words that means that the change in velocity achievable is equal to the effective exhaust velocity times the natural log of the initial mass divided by the final mass.  So we can see that the greater the ratio between the initial and final mass of the rocket, the more effective the rocket can be.

In the below picture are depictions of two rockets.  The one on the left is a single stage rocket. The one on the left is a multi-stage rocket.

Now lets look at the two rockets halfway through their flight.  The single stage has a lot of empty fuel tank mass that we are having to carry with us.  The multi-stage has dropped its empty fuel tank and become a smaller, leaner, more effective rocket.  

The hardest a rocket has to work is immediately at launch as it is trying to lift the most mass (all that unburned fuel) and doing so in the greatest gravitational environment (closest to Earth) and in the thickest atmosphere.  That means a bigger engine is needed at launch than higher up.  While the single stage has to keep using that oversized engine, the multi-stage can drop that big heavy engine and start using one designed for vacuum only.

No, solid boosters are not essential.  Solid boosters are very effective, but many rockets do not use them.  For example, the most powerful rocket ever used was the Saturn V that took us to the moon.  It did not use solid boosters.

SOURCE QUORA