I've never tried it, but there's an open source desktop / browser-based visualisation suite called WEAVE (short for Web-based Analysis and Visualization Environment). Like Tableau, it's intended to let you explore data through an interactive click-based interface. Unlike Tableau, it's open source: you can download the source code and install your own version on your own machine which can be as private or as public as you want it to be. Don't expect anything nearly as slick and user-friendly as Tableau, but it looks like an interesting, powerful project for someone prepared to put the time in to learning to use it.
If you're interested in the web programming option, here's a slightly more detailed write-up I wrote on Raphael and D3 for stackoverflow.
There are also some free (not open source) online datavis suites worth mentioning (probably not suitable for direct DB connection but worth a look):
- Raw by Density Design - blog introduction - (hit "Choose a data sample" to try it out) - mostly copy and paste based, not sure if it has an API that can connect to a database but good for trying things out quickly.
- Tableau Public - a free-to-use online version of Tableau. The catch is, the data you enter into it and any visualisations you create must be publicly available.
WEAVE is the best GUI-based open-source tool I know of for personal visual analysis.
The other tools listed are top of the range tools for online publishing of visualisations (for example, D3 is used by and developed by the award-winning NY Times graphics team), and are more often used for visualisation in the context of public-facing communications than exploratory analysis, but they can be used for analysis too.