instead of using matrices, we can use quaternions to describe orientations. It's quite important to be aware that a quaternion just represents a mirroring along an axis, and therefore cannot represent a scale or translation components. That's why you see Quaternions decomposed into TRS rather than just Quaternions.
Some people think that quaternions represent rotations. They don't. They represent mirroring. The mirroring can be used to effect a rotation, but it's a side effect, not a natural capacity.
The s,i,j,k values tell you in what way the plane of mirroring has been manipulated. values in i,j,k tell you how much the mirror rotates about each x,y,z axis. This is a simultaneous rotation though and is difficult to mentally represent. If you imagine a 2D version of the quaternion, you can see that with no rotation, pre-mirroring, then mirroring with the same mirror gets you back to where you started. If you rotate 180 degrees, the same will happen because the mirror is still effectively the same angle as before. If, however, you rotate by 90 degrees, you get a 180degree rotation back because you are mirroring in two different axes (which is effectively the same as a 180 degree rotation).
For general use however, you can assume that a quaternion is a rotation value. One very useful benefit of using quaternions is that you can interpolate between two of them without losing the fact that the transform is a rotation. When you linear interpolate a matrix, you are just interpolating basis vectors and an offset. When you interpolate quaternions, you are effectively interpolating a rotation. This is used in animation, both hierarchical (for characters) and camera (for swinging sweeps), and can also be used to define the rotation of entities in the world (so they can be manipulated freeer and easier).