Highcharts is my fav and default charting library.

http://www.highcharts.com

Highcharts is not open source but is free to use for non-commercial use. I have paid the $80 to use it on commercial websites and client/managers are always very impressed with its capabilities. I think its well-worth the $$

http://www.highcharts.com/license

Highcharts is open source, but it has a restrictive license for commercial use. It is free for non-commercial use (personal, non-profit) and it is also free during development, even commercial apps.

I also highly recommend. It's a pleasure to work with. Sites using it include MixPanel if you want to see it in action.

http://www.highcharts.com/license

There's always Protovis: http://vis.stanford.edu/protovis/

Protovis is fantastic. It goes well beyond charts and graphing, but you can create some wonderful, interactive charts with it.

There's flot: http://code.google.com/p/flot/ Raphael.js (along with gRaphael): http://raphaeljs.com/ jqPlot: http://www.jqplot.com/

I'd recommend flot out of these three, had the best experience with that one so far.

Check out Flot for jQuery: http://code.google.com/p/flot/

Some excellent examples here (scroll down):

http://code.google.com/p/flot/wiki/FlotUsage

For time-series, I like dygraphs: http://dygraphs.com/

For server-side chart generation, I tend to use matplotlib (semi-recently changed from gnuplot).

One option, depending on the type of chart your looking for, might be the Javascript InfoVis Toolkit: http://thejit.org/