A Stick catch is used when the flyer finishes their trick facing upwards. In this position, the flyer and the catch's hands are facing the same way, so they cannot grip each other (so a Wrists Catch is not possible). Instead, the catcher holds a short bar, known as a 'Stick' which the flyer grabs instead.
One main difference is that the flyer must grab the stick themselves, although the catcher can place it into their hands for complex tricks. Many catches on a bar to bar rig are also possible with a Stick (but much less common), so if you are looking for inspiration you can look at the various Bar Catches in this section.
Examples of tricks that use this type of catch are: Reverse Knee Hang, all Reverse Position tricks, Cutaways with whole numbers of twists (zero, full, double full etc)
You can find out more about this trick in my books:
IMAGE CREDIT: Jon, Austin and the team at Revolution Trapeze