How does bombarding a molecule with electrons cause ionization? I am a chemistry student, and my current are of study is mass spectrometry. 

In the ionization process, atoms are bombarded with electrons, but how does this work? I assume the flow of electrons attract the electrons of the atom, thereby removing them, but where do the atom's electrons go?
This type of ionization is known as electron ionization (there are many types of ionization methods used for MS). One possible reaction for EI is:
$$\ce{M +e- -&gt;M^{•+} +2e-}$$ source

If the incoming electron has the correct energy to interact and passes close enough to the molecule, it's electric field can impart enough energy to one of the molecule's electrons for it to be ejected. What happens to the electrons after that isn't particularly important as far as ionization is concerned, but in most EI sources, electrons are emitted perpendicular to the axis of the sample source and are collected on the other side by an electrode, preventing charging of the source chamber and focusing the emitted electrons into a beam.