1) Euler angles<->matrix<->quaternion, in 3D, they are all used to describe the orientation of primitives.
2) angles: 3-tuple,quat: 4-tuple,mat 9-tuple, these are memory footprints
3) compared to other two formats, quat has variant interpolation methods for smoothing the rotation
quat was invented by Dr. Hemillton and was firstly introduced into computer graphics applications by Shoemake.
For complete understanding, please refer to linear algebra textbook