If you are wondering how to capture targets alive in Red Dead Redemption, it is not that hard at all. To capture targets alive, you will need to do the horse mission from Bonnie and get the lasso.
Once have the lasso rope , aim and throw the lasso, hold on to both the aim and fire buttons, to pull him near you, till you will see a new tie up option and then a cutscene.
If you are on top of horse, aim with lasso, then fire and hold on to both aim and fire. With that press Press Y button in xbox (triangle for ps3) to get down from the horse go near him. Once you go near him you will see a new option to tieup and then cutscene.
If you are in capture bounty misson, you will need to carry the target to your horse and load him up. Thats it.
Without the lasso, capturing alive wont work even if you shoot him on the leg or shoulder.
Good luck and enjoy this game!