Sounds like they're the gamedev equivalent of a Monad? Which I only vaguely understand as a result of a poster on this forum's "Everyone explains it wrong" style post. Hoping you get a similar response with this.
Basic concept, 1 dimensional form: you want to represent a direction in a 2D plane. You can use one number, a heading angle, but at some point you reach a full circle and the number has to wrap. This creates annoying special cases. So another approach is to use a 2D vector, a point on a circle. Those are usually normalized so that x^2 + y^2 = 1. No angle is "special". You can average and filter such vectors without problems, for example. This is called a homogeneous representation, because it behaves the same everywhere in its space.
Now upgrade to 2D - latitude and longitude. Near the poles, small positional changes cause huge latitude changes, and computation error increases. This is a serious problem in navigational systems. So it's common to represent latitude and longitude inside of GPS systems as a 3-component vector, a point on a sphere, in what's called "Earth-centered, earth fixed" form. Now you can average or difference measurements without special cases. (Yeah, WGS-84 to compensate for planet not being a perfect sphere, etc.)
Now upgrade to 3D orientation. That's a quaternion. It's a point on a 4-dimensional hypersphere. This can be mapped to a 3D vector pointing in space and a roll around that vector, or to pitch-roll-yaw, etc. As above, a quaternion is a unit vector. It's hard to visualize this, so just shut up and calculate.
I am going to be the HN pedant here, and for that I apologize.
Quaternions don't specifically need to be unit vectors or points on a 4-D hypersphere. They are any number with 3 imaginary parts and 1 real part. Being a unit operator is a property of a rotation (this holds true even if you aren't using quaternions or if you extend to n-dimensional rotations), not a property of quaternions.
In the context of game development, "quaternion" almost always refers to a normalized one that represents a rotation. Many engines aren't even capable of representing anything else with their quaternion types and enforce this automatically.
That's probably because pretty much all uses of quaternions in games are for rotations, which involve normalized vectors. As a mathematical concept, quatetnions are far broader and more applicable than just rotations. This is a generic mathematical concept, not something specific to graphics.
Wow, I had never seen this explanation before. That makes so much sense! Now I finally understand why we always normalize the quaternions and why that is a sensible operation...
