Yes, having a bug tracking system helps immensely when you get a lot of issues seeing as you automatically get information and communication categorized to the relevant issues and can track progress without having to rely on people formatting properly. If you get a lot of bugs it also helps a lot to have a system in place for grouping them into categories and assigning responsibilities.
TRAC is a free system that works very well, though it is more along the lines of a project management utility that includes an issue tracker.
Roundup is another, pure issue tracker that is slightly simpler.
As for commercial solutions, We use
Hansoft at work and while that is very good it is an all-encompassing production management tool that might be overkill for a project with only two developers. We've also used
Devtrack and
Testtrack which were both very solid solutions with a lot of options for customization, but again, they might be overly complex for a small company.