I'm starting to learn more about quaternions and its geometric meaning in the world of computer graphics.

From reading a borrowed book late in the night, I recall that one of its geometric meaning is that it can rotate a model once by an arbitrary axis spherically.

Can it allow affine transformation, so that a vertex Vn in an object space be rotated by a point (x + O1, y + O2, z + O3), where (O1, O2, O3) is the object space origin, and Vn is one of the vertices in a vertex buffer object?

Unless I have to re-borrow the book again...

Can it allow affine transformation, so that a vertex Vn in an object space be rotated by a point (x + O1, y + O2, z + O3), where (O1, O2, O3) is the object space origin, and Vn is one of the vertices in a vertex buffer object?

With quaternions rotation and scaling (and hence mirroring) can be expressed, but not translation (AFAIK). The usual case is rotation what requires the quaternion to be of unit length.

It is possible to rotate a position vector, as is described e.g. here. As with other rotation representations, the axis' of rotation direction is encoded in the rotation, but passes through the current (0,0,0) and hence may need to be translated explicitly if required.

As haegarr said, a unit quaternion represents a rotation in the vector space R^3, but it cannot represent a translation. You can represent a "movement" (which I define as an affine transformation that preserves distances and orientation) as the composition of a translation and a rotation, and the quaternion is a fine way to encode the rotation.