I would like to know why particles follow a circular path in a magnetic field, what path they trace and how this path is measured and controlled?
The motion of a charged particle in a magnetic field is the manifestation of the fundamental relationship between magnetism and electrostatic effects. Exactly why the fundamental forces are the way they are is still beyond modern day physics which is why it is hard to give a satisfactory answer to the first part of your question. It just has to be accepted that magnetism and charge are intrinsically linked without knowing the reason.
How the particle behaves in the field however is very well understood and the basis of tools such as the cyclotron. The force exerted on the particle as a result of the magnetic field is given by:
Due to the fact that the vectors of the magnetic field, the motion of the particle and the force experienced by the particle are all perpendicular to one another, the force is directed at a right angle to the motion of the charge.
This means that the particle will follow a circular path (a spiral if it is not moving exactly perpendicular to the field). Varying the path is also quite simple as any adjustment to the strength or direction of the magnetic field will result in a change of the force exerted on the charge.