The ratio of the electron charge to its mass, e/m, can be determined from the curvature of the path of an electron in a magnetic field. That can be seen in a cathode ray tube (e.g an old tv) where the electrons are steered around with magnetic fields. The bigger the charge, the larger the magnetic force, giving more curvature. The bigger the mass, the less acceleration per force, giving smaller curvature.
The electron charge, e, can be measured by measuring the current produced by the flow of a known number of electrons. That can be obtained either from chemical reactions in batteries or from little single-electron solid-state devices, or even from vacuum tubes.
Putting together e and e/m lets you calculate m.
These are just a few of the types of measurements that can be used.
Whoops, I missed your second question. It's been answered several times on this site. One batch of answers appears here: http://van.physics.illinois.edu/qa/listing.php?id=1195http://van.physics.illinois.edu/qa/listing.php?id=1199
(published on 03/25/10)