I've recently been discharged from my job after a month, as- in my
supervisor's eyes- I wasn't processing the work quickly enough. She
had promised me intense training (with her) for at least the first two
weeks of employment, which never happened. She had another employee
answer questions I had (which were many, as the learning curve for
this position was huge); when this girl was able to help, she was
great; but this wasn't very often, as she had her own very demanding
position. After a month, I had managed to get to a point- learning
piecemeal- how to do just about everything my position entailed. I was
at a point where I could have really started speeding up, as I finally
knew everything I needed to do! My supervisor, however, said that she
just didn't see that I would be speeding up to the level they needed,
and let me go.
  I applied for unemployment, and indicated that I was fired/dischared
vs. laid off due to lack of work, as this was the case. All said &
done, my ex-boss & I left on decent terms. My question is this: don't
employers already pay for unemployment through taxes, thus not making
any difference to them if an employee applies for unemployment? (no
more money comes out of their pocket, does it?) I'm not sure whether
or not to expect my ex-employer to put up resistance to my making this
claim.

While it is true that employers pay unemployment tax, they do not all
pay the same percentage. Employers who discharge more employees are
taxed at a higher rate. One reason why an employer may oppose the
granting of unemployment benefits to a discharged employee is that the
employer's tax rate is based upon the number of former employees who
draw benefits.

unemployment benefits are paid when a individual becomes unemployed
through no fault of there own so I don't think being too slow
(unintentionally) is grounds for denial. Always file when un or under
employed. They will be getting in contact with you to find out why you
were discharged(probably after employer file their response, which is
time sensitive 10 or 14 days I think).