To experience magnetic force means that the particle is under action of force that is perpendicular to its velocity. The exact formula for the force is
\mathbf F_m = q\mathbf v \times \mathbf B
where ##q,\mathbf v## are the charge and velocity vector of the particle and ##\mathbf B## the magnetic field vector.
Here is how you can see the effect of magnetic force on electrons with your own eyes: take magnet and approach old TV or CRT monitor screen. Normally, the picture is formed by myriads of electrons flying from the hot cathode and falling on the screen. When you put magnet on the screen you will see that the colors get distorted by the magnet. The explanation is that the motion of the electrons and the magnetization of the screen is affected by the magnetic field of the magnet and hence they do not fall onto screen as intended for good picture.
In cyclotron, electrons move in horizontal circle thanks to constant vertical magnetic field ; magnetic force constantly curves the trajectory of the particle into a circle.
Yes, it depends on the frame of reference. So does the magnetic and electric field.
If you had two charges moving along with the same velocities, from your point of view, they would be surrounded by magnetic fields, but from their point of view, there is just electrostatic field and no magnetic field.
Electric and magnetic field are just two parts of description of interaction between bodies. "How much" of the interaction is described by electric force and how much by magnetic force depends on the frame of reference.