Why are electrons attracted to a magnetic field? In a cathode ray tube, the deflecting system can consist of either magnetic or electrically charged plates. I understand the fact that electrons are attracted or repelled by the electrically charged plates due to electrostatic forces. But why is it that the electrons are deflected by the magnetic plates? How does a magnetic field influence an electron?
 A: It is because of the Lorentz force, a basic law of electromagnetism. Its expression is
$\vec{F}=q(\vec{E}+\vec{v} \times \vec{B})$
What this means is that an electrostatic charge is only influenced by a magnetic field if it is moving (the first term is just the electrostatic force). The other is that the force is perpendicular to both the magnetic field and the velocity of the charge. Fore more details you can see here
A: 
Why are electrons attracted to a magnetic field?

Perhaps nobody knows.

You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird... So let's look at the bird and see what it's doing — that's what counts. I learned very early the difference between knowing the name of something and knowing something.

RICHARD FEYNMAN, The Physics Teacher, Vol. 7, issue 6
Your bird is named Lorentz-force.

why is it that the electrons are deflected by the magnetic plates?

I think this is the same question

How does a magnetic field influence an electron?

I think this is the same question

What are the quantifiable effects of a magnetic field on an electron?
Careful observation has produced a mathematical model of the phenomenon. See Wolfram jonny's answer for the equation that closely models how one seems to affect the other.
However it's not an explanation of why something occurs, just a mathematical description of what occurs.
It's a very useful model but it doesn't explain why something happens. It merely describes what happens.
What is the underlying mechanism by which a magnetic field affects the motion of an electron?
This would be interesting to know. I'm not sure if this is the question that you are really asking.
A: Moving charged particles are not attracted by magnet, but turned direction by them. This is what required in CRT -- to aim beam to different points. Also it is possible to build focusing devices of magnets: https://en.wikipedia.org/wiki/Magnetic_lens
A: This is an older question and an answer has been accepted but I'd like to add something to this topic. :)
When an electron moves, it generates a magnetic field.  So far as I know, this is how an electromagnet functions.  Electrons flow through the wire and a field is generated.
The same happens when electrons flow through a straight wire.  I once saw a physics lab demonstration where a wire was placed next to a magnet.  When power through the wire was switched on, the wire jumped dramatically.
This indicates that a magnetic field is generated by the electron movement through the wire.
Now here is the point.  If an electron's movement generates a magnetic field then the electron has become a magnet.
Since magnets exert a force on one another, when a small "magnet" such as an electron passes through the field of a larger, more powerful magnet, then we should expect that the electron will be deflected just as a small piece of magnetic metal would.
