How do magnetic fields affect electric charges? If a electron is placed in a fixed magnetic field, how will the electron get influenced by it? How does the magnetic field attracts/repulses the charge? 
 A: Magnetic fields do not attract or repel charges the way they do so for magnetic poles. They exert a force $q(\vec{v} \times \vec{B})$ on the charges.
The picture of an electric field and a magnetic field as separate entities is not entirely true. In reality, there is just an electromagnetic field which affects charges, currents, dipoles and such other entities. A simple illustration of this point could be - if you are in a frame of reference in which a charge is at rest, you measure only the electric field. However, an observer moving uniformly with respect to you, will see a magnetic field as well. And both pictures are true.
A: I assume you are conducting a thought experiment in which the electron is stationary in a perfect vacuum. In this case, in the equation from Amey Joshi, the electron's velocity is zero so the net force on it from the magnetic field is zero. The electron therefore feels no force and doesn't move.
However, this is an unstable equilibrium. The slightest motion of the electron causes a force at right-angles to the plane defined by the electron's velocity vector and the magnetic field vector (two lines at an angle define a plane) and so it ends up moving in a circle - assuming it doesn't escape the field.
There is also the magnetic moment of the electron to consider (it is a spinning electric charge so acts as a tiny bar magnet). This, however, should only cause the electron to align with the field and not to move linearly.
A: Electric and magnetic fields are unified into an electromagnetic field. The electromagnetic field is described in spacetime by the electromagnetic tensor.
Different observers (in different inertial reference frames) will see different combinations of the electric and magnetic components. When the charge is moving wrt to the magnetic field, there will be a force (Lorentz force) that will act upon the charge. An observer stationary wrt to the magnetic field will see only the magnetic field component that acts upon the charge. But from the charge's frame of reference, along with the magnetic field there is also an electric field component. So the complete equation for the Lorentz force that takes into account all the inertial frames of reference is:
$$\mathbf{F}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}$$
The transformation between inertial frames of reference is done using the Lorentz transformation.
