I want to give an answer for electrons which have both an intrinsic spin and a magnetic dipole moment. The key for the understanding why electrons in rest in relation to an external magnetic field get aligned and do not precess while moving non-paralle to the external magnetic field electrons undergo a precession is their kinetic energy in relation to the external field.
Electron in rest
An electron in rest when under the influence of a static magnetic field will get aligned with its magnetic dipole moment. For this case nothing more happens, there is no acceleration of the electron nor emission of EM radiation.
Electron in motion in relation to an external magnetic field
There is a well known macroscopic effect of the precession of a rotating wheel, called the gyroscopic effect. On the electrons level this effect can be explained as follows. Underthe influence of the external magnetic field the electron gets aligned with its magnetic dipole moment and by this a small amount of the kinetic energy gets converted into EM radiation. This is what we observe. Emitting photons the electron gets disaligned again. The photons momentum has to be compensated by the movement of the electron and by this the electron gets deflected from its straight trajectory. Once disaligned the game starts again and again. The resulting trajectory is not only a spiral path lasting until the electron is in rest with the external magnetic field and has exhausted its kinetic energy, but the trajectory in detail is made of tangerine slices.