A public tracker is a torrent tracker which doesn't require an account to use (e.g. The Pirate Bay, Kickass Torrents)--you just download the torrent file/magnet link and stick it into your BitTorrent client. Private trackers require accounts--even if you're given a torrent file from such a site, it won't actually work unless you have an account. Some of them can be elitist and exclusive, with byzantine interview processes that require you to jump through a lot of hoops to prove your worthiness. MyAnonamouse is on the other end of the spectrum; if you can read and answer a few questions, you can join.

The main advantages of private trackers are quality/consistency standards and availability of old torrents. With public trackers, there might be ten very similar torrents for the same thing, so you have to do some legwork to figure out what meets your needs the best. Private trackers will require some bare minimum amount of metadata and have standards to prevent true duplicates and allow better torrents to trump inferior ones. On public trackers, any seeding (continuing to upload a torrent after downloading) is done purely out of the goodness of one's heart, and so it's not uncommon for old torrents to have 0 seeders and be impossible to download. Private trackers require/incentivize seeding, so it's rare to have this problem.

Neither is superior to the other, but they can complement each other. For instance, I use public trackers for the vast majority of my anime needs, but if I need an old obscure anime or a well-organized music collection my private tracker rarely lets me down.

In this case it means you’ll have to pass a simple interview to use the site.