A quaternion is a set of 4 numbers, [x y z w], which represents rotations the following way:

// RotationAngle is in radians
x = RotationAxis.x * sin(RotationAngle / 2)
y = RotationAxis.y * sin(RotationAngle / 2)
z = RotationAxis.z * sin(RotationAngle / 2)
w = cos(RotationAngle / 2)

RotationAxis is, as its name implies, the axis around which you want to make your rotation.

RotationAngle is the angle of rotation around this axis.

So essentially quaternions store a rotation axis and a rotation angle, in a way that makes combining rotations easy.

Reading quaternion

The xyz components match roughly the rotation axis, and w is the acos of the rotation angle (divided by 2). For instance, imagine that you see the following values in the debugger: [ 0.7 0 0 0.7 ]. x=0.7, it’s bigger than y and z, so you know it’s mostly a rotation around the X axis; and 2*acos(0.7) = 1.59 radians, so it’s a rotation of 90°.

Similarly, [0 0 0 1] (w=1) means that angle = 2*acos(1) = 0, so this is a unit quaternion, which makes no rotation at all.

refer to: http://www.opengl-tutorial.org/intermediate-tutorials/tutorial-17-quaternions/