What is Fullerene?

To understand Fullerene, let's start with the difference between Fullerene and Graphite and finally, we will define Fullerene.
In Graphite, carbon atoms form layers with a hexagonal structure, and each carbon atom is connected to three carbon atoms ( so having sp2 hybridization). Still, Fullerene having fused rings of carbon atoms(e.g. C60, C20, C70 etc.) well these having mixed state of hybridization depending on bonding.
A diamond which is another famous allotrope unlike Fullerene here each carbon atom is connected to another four carbon atoms( so having sp3 hybridization).
So, they are structurally different, which clarifies the definition of allotropes here. Also, we have seen they are having different types of hybridization depending on bonding.
Now we are in an excellent state to define Fullerene. Fullerene is the allotrope of Carbon having fused rings of carbon atoms connected by single and double bonds( leading to mixed hybridization) forming a closed( e.g. hollow sphere, ellipsoid) or partially closed cage (e.g. cylinder).