You’ll probably be flying into Tahiti on Air Tahiti Nui, which is an international airline that flies huge and comfortable jumbo jets, but then you’ll change planes to board Air Tahiti, which is the domestic airline of turbo-prop jets, and is not affiliated with the international one. Obviously, they work together with their schedules, so if the long-haul flight is a bit late the island planes will wait for them, but still you’ll need to get your luggage and go through Customs and Immigration in Tahiti anyway.
There are about 7 flights per day from Tahiti to Bora Bora, starting at 6:45am and the last one going out at 5:10pm. Most go nonstop to Bora Bora, but a few of them stop off at another island for 15 minutes to drop off and pick up passengers. A nonstop flight will take about 45 minutes, and one that stops will take about 1 hour and 20 minutes altogether.
Again, you can book your entire journey for one fare, and that will choose the most efficient connecting flight for you, or perhaps give you options of connecting flights, with longer layovers usually saving you a bit of money.
You can also book the Air Tahiti flights in person once you arrive. The prices are actually the same either way, so you can book separately on the Air Tahiti website as well. These inter-island services all over the world usually operate like buses or ferries in that locals use them frequently for a variety of things, so there’s almost always a few seats available at the last minute, and even if this one is full then the next one an hour later will have seats.