At the time of writing in March 2013, bbPress 2.2.4 requires a Linux server with the following to run:
PHP Version 5.2.4 or higher
MySQL Version 5.0 or higher
It is also recommended that a server has Apache with mod_rewrite installed; however, this is not a necessity. Don't worry too much about these requirements as an e-mail to your web host, or any potential web host should give you this information. However, many Linux-based web hosts run these services as standard.
At the time of writing, the requirements for bbPress are exactly the same as WordPress, so if you are running the latest version of WordPress, you can run bbPress.