Disconnect the power
Start by turning off the breaker, and pulling the serviceman disconnect, which will typically look something like this.
This will insure no electricity is flowing to the condenser unit while you're working.
Open the unit
Next you'll want to disassemble the unit, to allow access to the electrical parts. This will vary from unit to unit, so check the owners manual for the procedure for your unit. Once you have the unit opened up, make sure to discharge the capacitors.
These things store enough power to kill you, so you don't want them to discharge accidentally.
Resistance is not futile
Once the power is completely removed from the unit, it's safe to start poking around (electrically speaking, don't go busting the refrigerant lines). Start by tracing the wires from the condenser fan motor, back to where they connect in the electrical box. There should be 3 or 4 wires. In my unit, I had Black, White, Brown, and Brown with a White stripe (your model may vary). To determine if the motor is good, you'll measure the resistance across each coil. To do this, you'll have to disconnect the wires, so the motor is no longer part of the circuit (make note of where the wires connected).
Typically you'll have 3 wires, start, run, and common (we'll ignore my 4th wire in this answer). Set your multimeter to measure Ohms, and start measuring. You're going to measure the resistance between each combination of two wires to determine what each wire is, and if the motor is still good. Let's start with Black and White...
Black -> White = 15.9
Black -> Brown = 35.4
Brown -> White = 51.2
Knowing that...
Common -> Run = Lowest resistance
Common -> Start = Medium resistance
Start -> Run = Highest resistance
We can determine that...
Black = Common
White = Run
Brown = Start
If we also know that the two lower readings should always add up to the larger reading, we can safely say this motor is still good. If you measure 0 or infinity between any pair, that means you have a shorted or an open winding and the motor should be replaced.
Repeat the same procedure for the compressor motor.
Shorts on the ground
The other thing you'll want to check for, is shorts to ground. Set your multimeter up to test impedance. Put one probe on the equipment grounding conductor of the feeder, and the use the other to find a solid ground on the motor. You may have to scratch some of the paint off, especially on the compressor. Once you've found a solid ground, measure from each motor wire to your ground spot. If the meter beeps or give a low resistance reading, you have a short to ground. As with the resistance test above, the motor should be isolated from the circuit when doing this test (once a solid ground is located).
It seems to me, after reading all the stuff in your first question and on here that the problem is inside your cloths dryer. There is a good chance that some part of the heater coil is becoming shorted out and causing huge amounts of current to flow when the coil becomes hot.
The probable reason that it is the main breaker that trips is due to that breaker being the most sensitive to the sudden current surge short that occurs in the dryer. It may be an indication that that main breaker needs replacement but before even going down that road it is time to unplug the dryer and do a through investigation of its innards, particularly its heater unit and the wiring thereof.
Best Answer
Since neither the washer nor the dryer breaker are tripped, yet the main did trip and power to those circuits cannot be restored by resetting the main, I suspect either the circuit breakers, busbar section, or both have become damaged inside the panel -- this means the next steps involve turning the main breaker off and disassembling the branch circuit panel, looking for damage to the busbar tabs and breaker contact clips, as well as the breakers themselves.
If you're not comfortable doing this yourself, now's the time to get an electrician in; while you have them in, have them replace the misfit breaker with one of the correct type -- i.e. a THQL120 instead of the BR120 that was improperly fitted there, and install a blank over the gaping hole in the main panel.
(It's likely that the misfit breaker was making poor contact in the first place and just quit making contact with the busbar during whatever caused the main to trip. Replace it with a THQL120 and things should be good.)