What are quaternions? Let's start with what quaternions once were: an attempt to extend Complex numbers to three dimensions.
Back when the geometric interpretation of complex numbers as a plane was reasonably fresh, Sir William Hamilton became interested in finding a system of algebra that would allow him to express three dimensional space in the same way. To do this, Hamilton needed a way to add and multiply points in 3 dimensional space together.
Addition came easy. Picking some arbitrary origin and axes 1, i, and j to work with, Hamilton just defined (a + bi +cj) + (d + ei +fj ) = ( a + d ) + ( b + e )i + ( c + f )j.
Multiplication, though, was a problem. Assuming that these new quantities were distributive, Hamilton needed to define ij in such a way that various other properties still held. Despite his best efforts, he couldn't do it.