There are many ways to propagate plants. One of the easiest and most rewarding methods is called air-layering. Air-layering is simply getting a branch to grow roots from some place on the branch. This is done by scraping off a section of the cambium layer (the green layer under the bark) all the way around the branch, brushing the exposed white sapwood with root tone, surrounding the entire scraped off part with a handful of moist peat moss (or other rooting medium) and then enclosing it in a waterproof barrier such as plastic wrap. The bad news is it's likely too late in the season to try this outside now. It will work on just about any woody houseplant any time of the year.
Here is a Lion's Head Maple (Acer palmatum 'Shishigashira') which was successfully rooted using the air-layering method. I severed it from the mother plant and potted up into a one-gallon pot. The roots are still young so I'm keeping it in a cool, shady spot until it's more established.
Everything I needed to know about air-layering I learned online. Here are two videos I'd recommend watching if you're interested in trying this out for yourself:
Dan Hinkley - Air Layering
Mikes Backyard - Air Layering