To fully understand an inverse spinel structure, one must understand a normal spinel structure. A normal spinel structure has the oxygen atoms closely packed together and the ions with a three-plus charge form an octahedral (eight-sided figure) and the two-plus charged ions form a tetrahedral (a figure with four triangular faces) that connects to the oxygen atoms.
As the name implies, an inverse spinel structure is effectively the opposite of a normal spinel structure. Instead of the three-plus charged ions occupying an octahedral, they instead flip position with the two-plus charged ions in the tetrahedral.
The structure of an inverse spinel occurs primarily because certain types of ions with a positive two charge naturally prefer to be in an octahedral and they force the positive-three charge ions out. One example of an inverse spinel structure is Mg2TiO4, while the most common, normal, spinel structure is MgAl2O4.