What are the rules surrounding when to make payroll tax payments for small businesses with 1 or 2 employees?
It depends on the size of the payroll, not on the number of employees. Probably you need to file Form 941 quarterly under this scenario. You may or may not need to deposit taxes more frequently. If you must deposit, then you need to do it electronically.
I excerpted this from the instructions for Form 941:
I would say that probably for two employees, you need to deposit by the 15th of each month for the prior month, but you really need to check the limits above and the deposit schedule in Pub 15 (as referenced above) based on your actual payroll size.
Note that if you have a requirement to deposit, that must be done either through EFTPS or by wire-transfer. The former is free but requires registration in advance of your first payment (they snail-mail you a PIN that you need to log-in) and it requires that you get your payment in by the night before. The latter does not incur a charge from the IRS, but your bank will likely charge you a fee. You can do the wire-transfer on the due date, however, so it's handy if don't get into ETFPS in time.
This is all for federal. You may also need to deposit for your state, and then you'll need to check the state's rules.