What is this wooden puzzle called, and how do I solve it?
I agree with Gareth McCaughan's comment that it is unclear that such rope-type of disentanglement puzzles actually have names, because they are simply too common and too varied. I finally found an isomorphic version with the product name "Family Games Flabber Floovers Brain Teaser Puzzle (Gripper)", but I do not believe that the puzzle's original name is "Gripper".
In any case, there is a generic solution to most such puzzles. You simply have to not move any of the parts except the rope, by thinking how you would detach the rope if the pole was not there.
You would easily take the red-circled part and push it through the indicated hole and get it to the green position where you want to make it cross both strands of the rope attached to the wooden pieces, as shown by the yellow arrows. This would be trivial if the pole was not there. Since the pole is there, you need to cross one strand at a time. I will show you for the strand connected to the left wooden piece, and you can do the same for the other strand. The idea is that you simply must cross the pole on both sides of the wooden piece. To cross on the left side, push it through the pieces' holes and across the pole end and back out through the same holes, as shown by the blue arrow. To cross on the right side, push it through the right piece's hole and across the pole end and back out through the same hole, as shown by the pink arrow.