Quaternions provide a different way of representing a transform. The representation is limited to rotations, optionally you can try to use it for mirrorings too, but it get's a little awkward so we don't normally do that.

Matrices provide a complete solution to representation of the orientation and position of an object, so they can be considered to be the de-facto solution for static scene transform representation, however, because they are a linear re-combination, they do suffer when used dynamically.

When you create your model matrix, the one for you windmill, you would normally generate a matrix (something like Matrix::MakeXRot( bladeAngle )) then set it into the rendering transform hierarchy. All well and good. What this is doing is setting a static matrix every time you change the bladeAngle value. If you are rendering a robotic arm, you would have a number of different matrices at different levels in that hierarchy, each one concatenating with another to finally provide that finger transform.

If you're not provided with all the angles, and instead being told that you have a start and end value for each transform in the hierarchy, then that works fine for the examples so far, you interpolate the angles and produce new matrices that are in-between the start and end matrices. This works fine because we're still in the world of scalars, interpolating between 0 and 100 for the windmill just gives us a value that we use to generate a rotation matrix. So all is fine.

But, when we are given the matrices, and not the rotations, suddenly we have a surplus of information, and we're not sure what to do about it. If you have a character end position that is a 180 turn from its start position (a Y-axis turn of PI), then you cannot tell which way to turn to interpolate, nor can you tell whether or not the real intention was not something more obscure, like a scaling to -1 in both the x and z axes.

Okay, so that last possible animation is unlikely to be a feature, more likely to be a bug, but that's what happens if you linearly interpolate a matrix. That's what normally happens when you linearly interpolate a linear transform.

This is where quaternions come in: they aren't a linear representation of orientation, not only are they non-linear, they're also not really rotations. The best explanation I've read / figured out for myself about what they really are, is that they are a kind of 4D mirroring operation, which when applied twice gives you the rotation you would expect. As far as anyone using them should be concerned though, they are vectors that give you a space for doing interpolation of 3D rotations.

*A quaternion represents a rotation, it can be interpolated to produce a rotation that will be on the shortest path between the two given orientations. Quaternions can be interpolated linearly, spherically, or used in higher order functions such as beziers or TCB splines.*

That is what you need to know. It allows you to interpolate properly.

Spherical linear interpolation, which leads to very smooth results, can often be relegated to trying too hard territory. Most animators and motion capture data already provide enough keys that doing the interpolation that carefully is not really necessary. Don't forget that artists don't generally see sub frame data, so getting it right is pointless in most cases.

* Gimbal-lock is when you can't interpolate from one rotation to another without doing some long winded route all round the houses. It manifests in Eulers due to the polar representation. Any time you get near a pole, you have to work overtime to get the transform in the right orientation.