The Bohr model of the atom, consisting of a nucleus with electrons orbiting around it like a mini solar system, is no longer considered an accurate picture. Instead, quantum mechanics says that an electron in an atom doesn't have a definite position, but instead takes the form of a 'cloud of probability' covering the locations where the electron is likely to be found. This cloud is called an orbital, but the electron doesn't travel round the nucleus in the classical sense. That being said, the Bohr model is still taught below the undergraduate level because it shows the general structure of the atom.
Outside the atom, the motion of free electrons is most often caused by the electromagnetic force - electrons have charge -1, so are attracted to positive charges and repelled by negative ones. Kinetic energy can also be given to electrons via the weak nuclear force, which occurs in beta decay, and through direct absorption of energy from photons as seen in the photoelectric effect.
Most subatomic particles can move freely - the proton has charge +1, so feels the electromagnetic force in the opposite way to electrons. Neutrons do not have charge, but can be given kinetic energy in fission processes or by collisions with alpha particles, for example. Quarks, on the other hand, have charge, so can be moved by the electromagnetic force, but they also feel the strong nuclear force, whose magnitude increases with quark separation. After very short distances it becomes so strong that quarks can never be truly free - they are always confined inside larger particles called hadrons, the proton and neutron being two examples.