Why do we want gap in the core material while designing inductor?
Saturation is always an issue in both transformer and inductor design. If we're going to spend money on a heavy and expensive iron core, then we want to work it as near to saturation as we can.

The reason inductors are gapped, and transformers are not, is that they are trying to do different things. 

The purpose of an inductor is to store energy. This means that to get the core close to the saturation B field should take as much H field, that is ampere turns, as possible. This needs a high reluctance magnetic path.

The purpose of a transformer is to transmit energy, with as little stored in the transformer as possible. In fact, energy storage in a transformer is a Bad Thing, needing snubbers to protect inverter drives. This needs a low reluctance path, so no air gap, as as high a permeability as possible.

Here's an analogy I like to use, and it's a bit odd, so I'm cool if not too many people grok it, is mechanical energy. In this analogy, stress is the equivalent of B field, so the saturation level is equivalent to the breaking strain of a material. Strain, the elongation, the change in length, is equivalent to the H field, the ampere turns. The stiffness is equivalent therefore to the permeability. An air-gap is a rubber rope, that takes a lot of change in length to get up to a decent stress. An iron core is a polypropylene rope, which takes very little strain to get it up to stress.

Now, which rope would you use for a pulley system? Obviously the non-stretchy one. You don't want to store energy in the rope between the pulleys, you just want input to become output.

Which rope would you use to store energy? The rubber one. If both the poly rope and the rubber rope had the same breaking strain, you could store 100x the energy using the rubber rope, if it stretched 100x more than the poly rope.

Bonus marks. Why do we use iron at all in an inductor? It's to do with the magnitudes of the permeability, copper losses etc. It so happens that it's not easy for the current to 'get hold of' the air round a conductor. It's a long way round the conductor, the H field is very low for any given current. It needs a lot of current to get a decent field. That's equivalent to our rubber rope being very long and thin, so we need to use some poly rope to 'gear it down' to the sort of distances and forces that are more in keeping with the rest of our system. The iron core concentrates the H field down to the small air-gap.