A plate at one end is heated, which causes it to emit electrons which shoot off down the tube, several grids or plates going down the tube have a voltage applied to them which deflects the electrons flying down the tube which is what steers the beam, in more advanced tubes there are multiple coils to properly focus the beam via either electric fields or magnetic fields.
Is it a must that this be carried out in a vacuum? And just to be clear on what I'm thinking is the "beam-focuser" a coil in which the electrons past through?
Yes the focus elements are generally coils or magnetic chokes. It's very similar to optics, but free flying electrons are the photons and magnetic and electric fields are the lenses.
It'll occur in plain air, but in a much different way. The first thing you'll notice is the heated element will burst into flames in an oxygen atmosphere and all the electrons will collide with the air molecules around it deflecting it every which way among any number of other interactions such as high voltage arcing and any number of other fireworks.