To fully understand about Quaternions, you should first understand what Complex Numbers are. I'll explain what a Quaternion is in a general matter first, and then explain it's basics.
What is it?
A Quaternion represents a rotation of a point in 3D space around an axis. That's it.
Quaternion has it's name because it's formed by 4 values (quadruple). Don't worry why it uses 4 values now. Just know that each of the values need to be build using sin/cos to actually means anything.
If you have a point in space, and you want to rotate it around any axis, you can use a Quaternion to do so. You can use other techniques, if you want too. Quaternions are just another option. Depending on your problem, it might better suit your solution.