Quaternions tend to be over-complicated.
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.