What some unit testing tools display when all unit tests run
successfully.  For example, here is JavaUnit's GUI after a successful test run:

http://c2.com/wiki/junit.gif

The term "green bar" can also be a reference to
the wide paper with alternating horizontal green and white stripes that we used to get our printouts on. (ImissGreenBar)

See also: GreenBarPolicy and RedBar (the opposite of a GreenBar).
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Before you do anything risky with code, you want to get yourself to a green bar, or a known state.  When you're working under a green bar, with good tests, you can change code around, and improve it, more easily than when you're working in an unknown state.

At work we sometimes say we're "yellow-diffing a green bar."  Our check-in script does color syntax highlighting of our diffs, and the subtractions from code show up in yellow.  If you have a check-in with a net reduction in code, it's a yellow diff.  If you yellow-diff a green bar, you've managed to make code smaller without losing any fuctionality.  Obviously, you never want someone else to yellow-diff your green bar, so once you get to a green bar, you immediately look to refactor your code.

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In the old days of line printers and continuous pin-feed (that is IBM mainframes and the like), the paper had alternating green and white horizontal sections, and was referred to as GreenBar paper. Typical size was 14 1/2 inches wide, 11 inches high.

It is still available, for example:

http://www.business-supply.com/dept/2005280/sparco-1-2-green-bar-computer-paper.html

with a small picture.