When an enemy is "downed" they have taken too much damage and fall to their knees/crawl away. You will need to use a variety of weapons to get all 11 executions, some having two different ones depending on the button used.
1. Torque Bow - "Y" button
2. Torque Bow - "B" button
3. Longshot - "Y" button
4. Longshot - "B" button
5. Boltok Pistol - "Y" button
6. Shotgun - "B" button
7. Neck Snap - "A" to pick them up, then "X"
8. Curb Stomp - "X" button (doesn't matter which gun)
9. 1-2-3 Punch - "Y" button (any gun that is not on this list)
10. Boomshield - "Y" button
11. Lancer - "B" button
Credit goes to The Pants Party who posted this guide on xbox360achievements.org.