Being a freelance developer for the last 10 years or so, I have seen and used almost every project management software you can think of, open-source and closed, and I have found that the "best bug tracking tool" completely depends on your process. I employ and direct teams in an Agile process that involves loose story-based requirements with point-based estimates, two-week cycles of iterative development, planning and retrospective sessions, and look-ahead and show-and-tell meetings with the stakeholders. It's important to me, then, that my tools have a method of capturing all those pieces with as much detail as I need but no more.
On projects, I've successfully used Pivotal Tracker, Unfuddle, Redmine, Codebase HQ, Basecamp, Trello, and many others. I usually recommend people to Trello for light, agile management; it's essentially a digitized version of sticky notes and swim lanes. If your team actually needs a full-featured ticketing system, see Redmine (Rails), Trac (Python), or Mantis (PHP), depending on your language preference. Hosting a version of these yourself is fairly trivial, and numerous options exist for cheap or free hosted versions. For something in the middle, Github Issues is a good fit and the open-source clone GitLab.org replicates most of those features nicely.
However, if you're looking for a hosted version, you're probably looking to offload that tricky "backup" thingy, and in that case, how important is your data? How proprietary? What's your business model? If you're working on an open-source project, Github will give your team a free account with private repos, issues, wikis, and the like. If you're okay with your project being "readable by all", public projects on Pivotal are still free.
I'm mostly a consultant these days, leading small groups of junior or intermediate developers into a more productive, more mature, fully operational teams of senior software developers. One of the first things I teach folks is how to use a project management system... and why! It'll save your bacon if it's simple, effective, and reliable. With a few minutes of discussion about your project, I can probably help you select the right tool and service for your team. Let me know if I can help with that. Best of luck!