Archery is no easy game, combining dozens of tiny factors. A mastery of it, however, is likely to bring in a good deal of gilders.
Arrows are free to practice with, but cost 1G/10 arrows if you wish to claim rewards (If you are just practicing, do not press the stack of coins at the end of your session. If you want to claim rewards, when you are done, simply click the stack of coins and wait for your gilder count to be updated). To get arrows, click on the row of layed out arrows (horizontal). The other stack of arrows (vertical), with the number under it, shows how many arrows you have left.
To fire an arrow, Position the bow at the correct angle and position, and be sure you are taking wind into account. Then, press the left mouse button down. All changes in wind speed will cease. To put power into your bow, shake the mouse back and forth (while the button is still pressed down) until the bar across the top of the view is at the desired power. When you wish to fire, stop pressing the mouse button.
The result of your shot will be marked with a red dot. If you hit the target area of one of the posts, you will get a reward. The rewards are as follows (in order of closest to furthest post): 0.1G, 1G, 1G, 3G, 4G, 10G. If you hit the bullseye (the small, faintly red dot in the center of each target area), you will get an additional reward (in order of closest to furthest post): 2G, 3G, 4G, 3G, 5G, 100G. Your rewards will show up as points until you cash in, at which point they will be converted to gilders.