I am often asked which forum software I prefer and why.
I have had experience with most out there paid and freeware and have tested them on my servers for the speed and server load. To decide what is best for you will have to look at what you need it for. Some offer excellent open source access to the code so you can modify it and add your personal add-ons or some are designed to handle a huge amount of traffic.
For easy access and free download for use on a small site I would recommend the YaBB forum. I have run it myself and learnt allot from their coding. It is written in a friendly manner and easy to pickup.
Downsides: The perl version of YaBB runs off text files. This can put a huge strain on the server. I have seen fast servers take 100% CPU usage for seconds every time someone posts a message. This is because of all the text files it has to go through to complete the action.
Other options, for those of you who run large sites and would need a more friendly option then you can't beet vBulletin. It is hugely customizable with allot of features in the admin area. It is very easy on the server as it has very fast MySQL integration meaning you forum is run off a database. It is definitely well worth the prices.