In most AC condensing units both the compressor and the fan motor are capacitor run motors.
Capacitor run motors have two separate sets of windings, the "Run Winding" and the "start Winding".
The run winding is then connected directly to line 1 of the contactor. A second wire is connected from Line 1 of the contactor to one side of a capacitor. A second wire is connected from the second terminal on the capacitor to the Start winding of the motor. This configuration is putting the capacitor in series with the start winding.
We must remember that this is an Alternating Current circuit which means the voltage rises from zero to peak positive and back to zero, then to peak negative voltage and back to zero 60 times per second. As the voltage is rising the run winding is energized and the capacitor is charging. The moment the voltage changes polarity the run winding is not energized but the current stored in the capacitor instantly discharges into the start winding. The two windings are placed in the motor at approximately 90 deg out of phase in rotation so that when one winding is not energized to otehr one is, thus it produces a more balanced output.
If a capacitor run motor is run for even a short time without a capacitor the entire load is placed on the run winding which usually results in reduced RPM & motor overheating and failing.
Typically an OEM (original equipment manufacturer) fan motor will only have three wires, Common (White) Run(Black) & Capacitor(Brown) which is actually connected to the start winding.
When replacing an OEM motor with a universal replacement type we commonly find additional wires on the motor.
Not all AC condensers use the same RPM for the fan, and not all fans rotate the same direction so the universal replacement motors allow us to configure it to the needs of the unit.
The standard color code for a universal multispeed reversible motor is:
White = Common
Black =High Speed
Blue = Medium Speed
Red = Low Speed
Brown = Capacitor
Occassionally we find a second capacitor wire:
Brown with white Stripe = Capacitor common
Purple & Yellow for changing rotation.
In your case your replacement motor is a single speed motor so it does not have the blue or red wires. The black wire is the run winding.
For your motor the Black wire should be connected to line 1 of the contactor.
The white wire is connected to line 2 of the contactor.
If you were replacing with a multi speed motor you would chech the RPM of the original motor and select the color that corresponds to that speed on the replacement motor and connect that wire to line 1 of the contactor, then cut or tape off the two remaining wires.
When the AC has a dual capacitor the center terminal of the capacitor is the capacitor common and is connected to Line 1 of the contactor.
If you had a separate capacitor for the fan motor you would check the top of the capacitor for a red dot or an embossed dot near one of the capacitor terminals. That is the identified terminal and the Brown wire is connected to the identified terminal, then the brown with white stripe is connected to the opposit terminal on the capacitor. (If the capacitor has no identified terminal you then just connect one wire to each terminal.)
The Brown wire from the fan motor is then connected to the capacitor "Fan" terminal and the Brown with white stripe is connected to the center capacitor common terminal.
The HERM terminal is for the start winding of the "Hermetic Compressor"
Some universal motors also have two purple wires and two yellow wires which are connected pruple to purple and yellow to yellow. That is for clockwise rotation. If you need counter clockwise rotation you then disconnect the wires an connect one yellow to one purple, then the other yellow to the other purple to reverse rotation.