Free and Open Source Software (FOSS), is software produced primarily for social benefit, rather than profit. Over the last 30 years, different organizational structures have evolved to produce this software. Most of these organizations fall squarely in the category of "civil society", though they exist in a realm dominated by for-profit corporations.
The organizations differ in size, scope, and focus, but generally share these characteristics: engineering software, providing legal support for developers and issue a common license, holding meetings and providing some form of governance. Additionally, some foundations boards include members from large computer or software companies, and thus, they also function as adjunct organizations for these corporations.
There's a big list at Wikipedia.
This list is very USA and Europe centered. For this, I apologize.
Apache Software Foundation
Blender Foundation
Eclipse Foundation
FreeDesktop.org
Free Software Foundation
GNOME Foundation
Internet Systems Consortium
Internet Engineering Task Force
KDE Manifesto
Linux Foundation
Mozilla Foundation
Open Database Alliance
The Perl Foundation
Software in the Public Interest
The Summit Open Source Development Group
Ubuntu Community Governance
World Wide Web Consortium
VideoLan Organization
xcf
Xiph Foundation
X.org Foundation
FOSDEM
GUADEC, Akademy
OSCON (a trade show)
eParticipation
TechPresident
NetSquared
There's been some social science research about FOSS development as well, but most of those articles are inaccessible without membership. I'll link to them at some point. In the meantime, there is related material at Groklaw, Indymedia, Net-Time.