The archer pull-up is an advanced pull-up variation that involves keeping one arm straight while relying primarily on the opposite side to do the bulk of the pulling. You can think of this exercise almost as a one-arm pull-up with a self-assist.
I'd recommend a solid foundation in basic pull-ups before even attempting this variation. If you can't do 10 clean reps without struggling, you're probably not ready to try the archer pull-up.
How to Do It
Begin like you're doing a very wide pull-up, but bend only one of your arms as you pull your chin over the bar. Due to the asymmetrical nature of the movement, your torso will need to shift away from the extended arm as you pull, with the top position resembling an archer getting ready to fire an arrow.
The hand of your straight arm may need to open and roll over the bar at the top of the range of motion, depending on your wrist mobility. You may also need to extend your legs slightly to the side to counterbalance the movement.
I Can't Do It!
You can cheat a bit by allowing your secondary arm to bend slightly in order to make the exercise less difficult. Once you get to the top, you can then extend the arm fully and attempt a negative archer pull-up. Soon, you'll build enough strength and won't have to bend the secondary arm at all.
As with other pull-up variants, most people will find it easier to begin with the primary pulling arm in an underhand grip.
Give it a shot. It's harder than it looks!