In order to obtain this achievement, you will need the lasso earned after the 6th mission with Bonnie MacFarlane; "Wild Horse, Tamed Passions".
The mission order is:
1. Exodus in America
2. New Friends, Old Problems
3. Obstacles in Our Path
4. This Is Armadillo, USA
5. Women and Cattle
6. Wild Horse, Tamed Passions
**Thanks to DimitriRascalov for supplying the mission order!**
Once you have procured your lasso, you will need to find a wanted poster located outside of a Sheriff's/Lawman's office. (I used the one on MacFarlane ranch). **Note: Wanted posters show up on your map as a greyish/white skull** View the poster by pressing "B" and select the bounty by pressing "Y". Once the bounty mission is selected, you will see a red skull appear on your map, this is the wanted man you are trying to capture.
Follow the map to the marker and fight the kill the posse protecting your wanted man but DO NOT KILL THE WANTED MAN. Once you have taken care of your prey's posse, select your lasso, target the wanted man and rope 'em. **Note: to make this easier on yourself, switch the targeting in your options to normal or casual, this way you can lock on the bandit and make him easier to rope**
Once the man is roped, hold on to the left trigger, walk up to the roped man and press "Y" to hogtie him. It is then the simple matter of picking the man up, putting him on your horse and taking him back to town. You may have to deal with other posse members out for revenge as you take him back so be quick. Once you drop off the hogtied man to the Lawmen, the achievement will pop after a short cutscene of them walking him into the jail. There you have it, an easy 10 Gamerscore.
Let me know if I am missing anything.