Advice on secure, simple web forum software

[Nerius Landys]

One of my not-for-profit websites is increasing in popularity and
users are requesting forums.  I am going to add forums now, but I'm
not sure which route to take:

1. Install the forums software so it runs on my webserver.
2. Use some third-party forums website [with their advertising probably].

I much prefer #1 to avoid any advertising (which my website is
completely free of).

Packages currently installed on my system that may be relevant are:

apache-2.2.11_7
php5-5.2.10
<no databases, e.g. MySQL, are installed>

I would like to get some recommendations on which route to take.  If
I'm going to install forums to run directly on my server, the software
needs to be secure (as few bugs or loopholes, exploits as possible),
the same UNIX user will be running a couple of other small sites.
Also simplicity of design is preferred, feature bloat and dependencies
are bad.  I'm willing to install a database such as MySQL.