A private tracker is a website that has the same functionality as a public tracker but restricted behind an invite. The advantages of a private tracker are retention, quality control, selection, speed, security.
animebytes would be the best private tracker for anime.
They're just like public trackers, but private so you'll generally need an invite to get access to them. They generally are much better than the public trackers. Not sure about an anime specific one though.
Pros: Good selection, direct access to a few scene groups (scene groups upload mostly on private trackers and these are leaked to public ones systematically), a base level of trust that long time uploaders won't upload viruses, fast download speeds.
Cons: most have requirements on keeping your ratio high which is hard to do if you don't have good upload bandwidth, there may also be a requirement to seed torrents for 10 full days (total, not consecutive) if you can't get to a 1.0 ratio on a given torrent. Your account can be deleted for inactivity.
There are some sites with a freeleech tag which means that downloads tagged with that don't harm your ratio but seeding helps your ratio. In the one I use, you still have to seed for the 10 days. In my experience, because I don't have a seedbox or a good up connection, I rarely can successfully seed at all to build ratio so I am somewhat limited in what I can actually download (only torrents larger than 14 GB or a box set) without throwing my ratio in the toilet.
I could pay for a seedbox or a premium membership to fix this.
Private trackers are great since people tend to seed things for longer, also to note generally keeping your ratio up is required. Personally, I use a private tracker when I need to find something I just can't find anywhere else, but with that said for anime nyaa should be the only place you really need.