This has to be my favourite method for propagating begonias. From a single leaf you can end up with a dozen or more plants. It's a relatively basic method and you only need a few things:
Potting soil (already moist)
Pot (large enough to fit the leaf when laid on the soil surface)
Plastic material (or similar) to cover the pot
Make sure to take the leaf with some stem. Then you can get another plant from the stem as well. Take the knife and slice each of the main veins about 5cm (2") from the stem. These veins then branch into smaller ones which can also be sliced (but keep the leaf itself intact). Again I wouldn't have the slices along a vein closer than about 5cm. Dab each vein slice with some rooting hormone on the bottom of the leaf.
Now push the leaf down onto the surface of the potting soil. It may not lay flat especially with some of the more ornate varieties. If this happens you can use small rocks to hold the leaf down. Now cover the pot with plastic ensuring there is at least 2cm (about 1") of room above the leaf. Put it a place with bright but indirect light. Heat will speed the process.
Within a few weeks you should notice what looks like weird mutations growing at each sliced vein. These will put out leaves and become independent clones on of the original plant. I usually leave the new plants in the pot until they're reasonable large and look crowded. Then I split them up into individual pots.
Generally the new plants only put out a few leaves at a time. I have no idea why, but the ones that I took the photo of seem to have a dozen or more tiny little leaves ready to grow at the same time. You can see the pot is already filled with roots to the point they have grown up along the surface.
Although this form of propagation is common, many people don't seem to know about it. Hopefully I've given you something new to try. Good luck!