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Nanorobotics can be realized with a combination of a magnetic field and a passive robot. The robot is equal to a tweezer which is controlled by an external field. This field is generated by electromagnetic coils. Even the topic is described in the literature since the 2000s, it is hard to understand how such a system works exactly. My question is: what kind of coil is needed to generate a vector field which drives the robot? As far as i know from the physics course, a normal coil can either be on or off, but how to change the direction of the field?

The magnetic field is proportional to the current. A fixed geometry coil can generate a field that is positive or negative or zero based on how much current is driven through the coil and which direction the current goes. This is limited by how heat is removed from the system.

The exact shape of the field can be influenced using a magnetic circuit to guide the magnetic field.

Magnetic fields combine so two coils can create a combined field that is any combination (sum) of the individual fields. Add as many coils as you want.

Other types of control can be done by rapidly switching fields (perhaps just a single field) on and off rapidly. This is what is done for the magnetic levitation desk toys.

There are other things that can be done by using the magnetic field to transmit power, or combining mechanical or fluid dynamic effects on the motion of the nano-robot with the magnetic effect.

AC motors operate by generating a vector field. The field on the stator rotates, and the rotating field interacts with the field on the rotor (generated, or permanent magnets, or induced) to move the rotor.

The motor fields rotate by varying the current applied to the stator windings because magnetic field strength is proportional to current. The orientation of the coil/winding determines the direction of the field, and then the applied current essentially determines the magnitude along that axis.