Check out Team Headquarters at www.Entry.com. Hosted solution that is very powerful, full audit, security et al and can import/export to MS Project as needed.
The online project management tool I use and recommend to my clients is teamworkpm.net.
It is simple, straightforward and provides daily reports to keep you on focused and on track. Free for 60 days! Then a modest monthly subscription fee.
I like Basecamp too, but it's more of a workgroup management tool, not a project management tool. Looked at from a project manager's point of view, I like being able to assign, hand off and comment on tasks, estimate completion dates, see what's next, what your velocity is, etc. For that I've had great luck with PivotalTracker - it was designed to support agile software projects and uses some terminology from that world, but I've found it to be excellent for non-technical projects as well.
In addition to scores of others, I've used all the ones suggested here (except Daylite - poor reviews, costly). Always leaning toward free plans or shareware options, the only one from below I recommend is Podio.
It can take a while to get used to, but that's because it offers just about all the flexibility you ever think you might need, through the "app market" where you install modules that create additional functionality. Zoho products for me have always been buggy or unintuitive/inflexible (and I've been marketing online since the late '90s). I also want to reduce the number of services for which I pay a monthly fee, which nixes Basecamp for active businesses.
Because Podio became a bit much after I moved away from lots of team work to more individual work, I now use Producteev. Recent updates have made it easier and more full-featured to use. Before that, I briefly tried Asana. Again, free, and very cool ... but its funky interface confused me a little and ultimately turned me away.
There are many others I've used, and I also have a spreadsheet of a few options I had one of my admins compile for project managers best suited for Web designers. Would be happy to share if you're interested. The "best" one depends on what sort of business you are and what specific features you need. For me, it was frustrating to find one that fit exactly, as if you have several specific *needs*, it's a crap shoot.
There was one, too, that plugged right into Gmail and was an AWESOME and simple task manager... whose name I can't remember. If that sounds interesting to you, I can locate it again, too.
Basecamp is cost effective and simple and therefore efficient to use. The new version has had some useful functionality stripped out of it but it still works well when there are many stakeholders, files and communications all need to be in one organised place and a coherent plan is required.
Basecamp was ok, but the low number of projects at a reasonable cost makes cumbersome structuring necessary. I found it most effective to treat "projects" really as file folders, with multiple related projects within each folder.
I've used both Basecamp and Podio. They are both good. I tend to like Podio personally.
Basecamp. Affordable effective and easy to use. No gismos, no complicated stuff.
Just a tool to get the work done.
Zoho Projects and AceProject are affordable options that are quite good. I prefer Zoho as it integrates with other Zoho applications and seems to have better reporting capabilities.