vbulletin forum board program has been around for many years, it is considered one of the most stable, feature rich and matured php forum board program around. It is robust, scalable and is able to serve any small communities containing several hundreds of members right up to communities with several hundred thousands of members, and that is why most of the large community forum board uses vbulletin as their forum platform.
vBulletin is not an open source software but a paid license forum board program, you are required to purchase a license key in order to use the forum program. However, the license key is not that expensive or perhaps consider rather cheap for the value you can get from the program.
vBulletin is modular in design, it allows third party addons to be added as plugins to the core program, therefore extended the already feature rich forum board, there are free plugins as well as paid license plugins that you can add to the forum board.
If you are planning to start a forum community, vbulletin should be your choice of forum platform, as you will have less chances to worry about the program bugs and spending time fixing little things here and there where it is common for open source programs, you will have more time to focus on building your community to a success.