Is the washer's transmission broken?
Here is a list of possible issues:

Solution 1:
Drive Block or Bell
The drive block connects the washer transmission to the agitator. If the drive block is worn out, the transmission shaft will still move back and forth, but the agitator will only move slightly or will not move at all. Inspect the drive block for damage. If the drive block is worn out, stripped, or damaged, replace it.

Solution 2:
Drive Belt
Inspect the drive belt to determine if it is broken or if it is loose on the pulleys. If the drive belt is broken or loose, replace it.

Solution 3:
Drive Motor (MOST LIKELY NOT YOUR ISSUE)
The drive motor might be defective. However, the drive motor is rarely at fault for this symptom. Before replacing the drive motor, first check all of the more commonly defective washer components. If none of those components are defective, consider replacing the drive motor. To determine if the drive motor is defective, first ensure that the motor is getting power. If the drive motor hums but doesn’t run, is noisy, or does not turn freely, replace it.

Solution 4:
Door Lock (MOST LIKELY NOT YOUR ISSUE)
The door lock secures the washer door shut during operation. Once the washer door is closed, the washer can begin operating. If the door lock is defective, the washer door will not latch properly. The door lock can fail either mechanically or electrically. Inspect the door lock for damage. If the door lock is damaged or does not latch properly, replace it. To determine if the door lock has failed electrically, use a multimeter to test each of the door lock switches for continuity.

Solution 5:
Agitator Assembly
The agitator might be damaged or stripped away where it connects to the drive system inside. To determine if the agitator is stripped inside remove it from the washer and inspect for damage or wear.

Solution 6:
Transmission
The washer transmission might be worn out or defective. Transmissions have gears in them that can fail causing the washer to not spin. However, this is almost never the case. Before replacing the transmission, first check all of the other more commonly defective parts.

Here are a few diagrams: