Thanks for writing. Bougainvillea grow pretty well from 3- to 5-inch-long cuttings taken in summer. Pot them up in moist sand, peat moss, perilite, or vermiculite and keep them humid, but out of direct sun. They typically root in about 6 weeks. Learn more about taking cuttings here: http://www.bhg.com/gardening/yard/garden-care/starting-flowers-from-cutting/
If you already have a bougainvillea, layering it is also an effective method. Get more tips for layering here: http://www.bhg.com/gardening/yard/garden-care/make-more-plants-by-layering/
Thanks for writing. I hope this helps!
---Justin, Senior Garden Editor, BHG.com