Why were halftracks so common in WWII and why are they no longer used?

I can give a little bit information on the second part of the question - why they are no longer used.

Half-tracks have generally evolved into APCs (Armored Personnel Carriers) which in turn have evolved into IFVs (Infantry Fighting Vehicles).

What are the differences between halftracks, APCs and IFVs?

In contrast to halftracks; APCs are fully tracked, tend to be fully enclosed (armored top), and generally have thicker armor. Fully tracked APCs have the advantage that they are able to cover the same terrain that tanks are able to - halftracks are a compromise between wheeled and tracked vehicles so they don't have the same level of cross country mobility as a fully tracked vehicle although they have better on road performance. Being fully enclosed also helps address some vulnerabilities of halftracks - troops riding in them would be very vulnerable to artillery fire such as air bursts, strafing planes, elevated machine guns, grenades and molotov cocktails. The heavier armor offers a bit more protection against light anti-tank weapons although most APCs would not withstand a direct hit against a contemporary AT weapon. Interestingly enough, the first 'purpose built' APC was created during World War II. The Canadian Ram Kangaroo was a conversion of obsolete tanks and self-propelled howitzers. The main purpose of both APCs and halftracks are to provide mobility and protection for troops directly into or close to battle.

IFVs are the latest evolution of APCs. The Treaty of Conventional Armed Forces in Europe differentiates between the two by armament - APCs are armed with cannons of less than 20mm, while IFVs have cannons of greater than 20mm and sometimes an anti-tank missile. Essentially, IFVs have heavier armament, generally heavier armor, and unlike APCs are expected to participate directly in combat to support their infantry. The first IFV was the Soviet BMP-1 which carried a 73mm cannon and an anti-tank missile launcher. This was a big technological leap - while IFVs could not go toe to toe with a tank because of their thin armor, they could still defend themselves against one. The cannon also allows them to provide fire support for their infantry similar to what a tank could do. APCs and halftracks are also armed, but generally with a machine gun that was primarily meant for self defense. The modern US IFV is the M2 Bradley which was influenced by the BMP, and also has a heavy autocannon (25mm) along with TOW missiles.

So the short summary is that half-tracks have evolved into the modern day IFV, due to technological changes - anti-tank capability only became realistically possible with the advent of guided missiles, and also doctrinal changes - the mission of personnel carriers is not only to transport troops but also provide fire support.