Answer:
The force that a black hole exerts on its surroundings is of the same nature as the force that the Earth, or any other celestial body, exerts, which is the gravitational force. A black hole with a mass equal to that of the Sun creates around it a gravitational field which is exactly the same as that of the Sun. This means that, if a black hole of equal mass were placed in the place of the Sun, we here on Earth would not feel any difference in terms of orbital motion. Of course, we would notice the lack of sunlight, but we would continue to orbit without any problem around the same point.
The diameter of the Sun is approximately 1.4 million km while the diameter of a black hole of a solar mass is around 3 km. The so-called relativistic effects only make themselves felt significantly if we get too close to the black hole (in this case, if we are less than, say, 100km from the black hole).
Black holes, like other celestial bodies, respect the Laws of Physics and do not have any will of their own, as is often mistakenly assumed.