According to the General Relativists, "Gravitomagnetism" is that phenomenon such that spacetime geometry and curvature change due to the currents of mass-energy relative to other matter in the universe. The word draws connotation with respect to the magnetic field in electromagnetism arising from charges in motion. But, the major difference between the two fields is the so-called equivalence principle which can nullify a gravitational field by creating an accelerating frame of reference.
In classical mechanical terms, this would be equivalent to eliminating fictitious forces in non-inertial frames by referring them to inertial ones. It is often observed the similarity to the Lorentz force and the Coriolis force. Whereas the former is non-zero in an inertial frame, the latter vanishes there. This is because the definition of the Lorentz force uses the velocity defined in the inertial frames whereas the Coriolis force uses the relative force in a non-inertial one. The two are related by the centrifugal force which can convert one into the other.
Two phenomena that Gravitomagnetism claims to explain is the Lense-Thiring (LT) effect and the de Sitter (dS) precession. The claim is that LT precession of an orbiting gyroscope is due to the gravitomagnetic potential, whereas the dS precession results from the non-commutivity fo the non-aligned Lorentz transformation of special relativity and the gravitational acceleration causing spacetime curivature in general relativity. In order to do these phenomena full justice one must have recourse the the Kerr metric. Nothing could be farther from the truth.
The LT precession arises from the rotation of the central body, and is present even when the gyroscope is at rest. The precession at the poles is parallel to the rotation of the central body whereas it is antiparallel at the equator. This is what the gravitomagnetic potential seeks to demonstrate. This has given rise to the colorful, if meaningless, phrase the the rotation of the central body "drags" the metric along. At the poles, the metric tends to rotate with the central body whereas the "dragging" of the metric falls off at increasing readius at the equator. If the gyroscope is oriented perpendicular with respect to the equation, the side away from the central body is dragged less than that closer to the body so that the spin will precess in the opposite direction to the rotation of the body (L Schiff).
The dS precession is nothing but the relativistic gravitation analogue of the Biot-Savart effect where the creation of the magnetic field is due to the flow of charge. In the dS precession it is the flow of matter which causes the precession.
The vector potential governing the LT precession is analogous to that of a vortex,
2 pi A = GI/c^2 w/r,
where I is the moment of inertial of the central body rotation at an angular speed, w. In analogy with vortex motion the coefficient would be w S, the vortex strength, S being the area, times the height, h. The latter is replaced by Gm/c^2 in the gravitational case where the mass, m, comes from the moment of inertia, I= m r^2, neglecting numerical coefficients depending on the shape of the central body.
The velocity is the curl of the vector potential A,
v = curl A = GI/c^2 (w X r)/r^3.
And the curl of the velocity is the so-called gravitomagnetic field
H = curl v = GI/(c^2 r^3) {3(w e)e -w},
where e is the unit vector pointing in the radial direction. It is clear that H is 3D, so how can it be used to describe the rotation of a black-hole, in The Black Hole Paradigm by K Thorne, is indeed surprising taking note of the fact that black-holes are 2D objects.
Apart from a numerical constant, the dS precession has a gravitomagnetic potential given by
H = Gm/c^2 (v X r)/r^3,
which is essentially the Biot-Savart relation where a magnetic field, H, is induced by the flow of electric charge, v. Again, the power of the denominator is governed by the coefficient Gm/c^2. In the Biot-Savart relation, the power would be two, and it would reduced to an expression for the angular momentum, (v X r)/r^2 =w.
Neither general relativity, in general, nor gravitomagnetism, in particular, is necessary to understand these two types of precessional motion.