My first forum that I managed was my own invisionfree forum. It was free, had (has, zetaboards does too) a lively community, and I got my first taste of mods and scripts. I don't know if any of you used IF, but for those of you that used it in 2005-2007 you may remember RPG Inferno. Became pretty obsessed with it because it was banned due to some code copywrite issues. From there I downloaded nulled IPB, worked with mybb, vbulletin, and a couple others.
Phpbb was and still is the best open source forum software available. Other forums have their perks and specialties, but in general phpbb takes the cake. There are a multitude of extensions and styles, a huge community, active support...
Almost anything that you need done, for the most part, Phpbb can handle. I'm a web developer/administrator on the side, and when a client needs a forum developed, I always recommend Phpbb before anything. In some circumstances I try and recommend converting to Phpbb, but those are few and far between as most of my clients already use Phpbb, and I can recommend them an extension or two they wouldn't have thought of, or a new style, to spruce up their site
I've had to go with others of course, only because there isn't an extension for every need... But it's mostly there and I love it. And it's pretty fast too. You can make Phpbb look, feel, and act like any forum out there and in the end users won't notice a thing.
Vbulletin doesn't have the community it used to, at all. IPB is nice, but it's also complicated to modify, and expensive. Mybb is mostly up to Phpbb in my opinion, but there are less themes, modifications, a smaller community... Vanilla forums only had gamification going for it. That's it. Xenforo is a baby vbulletin that markets itself as responsive when most Phpbb these are, or can be made so if you know your way around css and html.
Tldr; phpbb is the best open source forum software. Period.