Open Flash Chart
Open Flash Chart, a Flash-based chart system (no surprises here), features a very powerful API, possibly more so than most other options (including commercial offerings!). It has an interesting approach to chart generation – it fetches chart data from the server; the object makes a seperate HTTP request to fetch the dataset.
Honorable mention: PHP/SWF charts
PHP/SWF charts isn’t entirely open source, but I’ve included it with an honourable mention given its popularity. It offers a free version, with almost all the features of the commercial offering ($45/domain). With PHP bindings, it’s possibly the easiest to get started with if you already use PHP in your application, and its sister project XML/SWF charts happily takes any other server-side language. It also has the most visually appealing demo charts.