History:
Odoo started as TinyERP in 2005 before changing it's name to OpenERP in 2008.  Then in 2014 with the addition of modules that fall outside of the traditional ERP model such as CMS and eCommerce, OpenERP S.A. changed it's name to Odoo to focus on their future beyond just ERP software.

Cost:
Full source code for Odoo is available on GitHub and released under the GNU AFFERO GENERAL PUBLIC LICENSE, Version 3 (AGPLv3).  So like most Opensource software you can download, install, modify and use the software freely.  The cost of the software is your time to learn and maintain the software, development costs to personalize the software and associated costs in hosting the software internally or publicly.

There are a few items that Odoo does not include for free:

1. Upgrades between major version numbers (i.e. 7.0 -> 8.0).  Odoo charges for their upgrade service, however the community also has a free opensource upgrade option called OpenUpgrade.
    
2. Installing add-on modules through the GUI interface.  Odoo's built-in module installation system stops working once you have more than two users on a system without an Odoo maintenance contract.  However you can install (-i) and update (-u) modules using the command line interface.
    
3. Hosting services.  Odoo provides for it's customers an online hosted version of Odoo.  Their SLA (Service Level Agreement) includes replying to bug reports, provide back-up and security services as well as migrate users to new versions (including early access to new unreleased features) at no additional charge.  The price is based on the number of users and modules that are installed.
   
   Recently Odoo has announced that their hosted solution will be free for customers with one or two users regardless of the numbers of modules installed. However you server will have a YourName.odoo.com web address.  Odoo will not point your domain (YourName.com) to their free hosted solution.
    
4. CRM website themes.  Odoo is developing 70 themes for the website and eCommerce (see CRM website themes below).

Communities:
Odoo has many distinct communities where you can find and give advice.  First Odoo hosts it's own forum and several email lists on various topics.  The OCA (Odoo Community Association) has their own website and email list (publicly searchable) as well.  Many Odoo users post tips and information on their personal blogs.  And at least two private forums exist on Odoo related topics: StackOverflow.com and developpez.net (French).  There is even an IRC chat room for Odoo as well as Linked-in, Google+ and a Meetup group.

Odoo also hosts a yearly OpenDays event with Sessions, Workshops and Training.  The event is free to attend (there is a nominal charge for catering if you choose to eat at the event or wish to attend the after hours parties) and there are optional training opportunities available for a fee.  You can watch the OpenDays 2014 event online: Day 1, Day 2 and Day 3. 

Documentation:
The documentation for Odoo varies from version to version because, on average, Odoo releases their software every 15 months.  In general, Odoo tends to focus more on new software features and improvements than on development of user documentation.  A compiled list of known community and Odoo released documentation is available on the Odoo forums.

Training:
Training can be obtained through one of the over 500 Odoo certified partners located in more than 100 countries.  These events, some free and provided electronically over the web, are posted on Odoo's event page.  Also from time to time there are events posted on the OCA's website.

Software:
As stated above, the source code is available on GitHub for versions 5.0 through the current master branch (future development branch).  Odoo also provides nightly builds of source, Windows, Debian/Ubuntu and RedHat install packages. This is also where you can find copies of TinyERP version 3.0 and 4.0 available.

If you wish to build your own packages, instructions are available for Odoo v8 as well as OpenERP v6.1 & 7.0 builds.  For Windows users there are also instructions on compiling an executable.

Modules:
Modules are the way you can add new functionality to the system and a way to only install functionality that you need.  They are written by Odoo, official Odoo Community partners and by individuals.  Odoo provides an online apps store that can be used to publicize the existence of an odoo app.  Currently there are over 2,000 apps for OpenERP 7.0 and the number of apps for Odoo 8.0 are just beginning to be added to the system.

In general, if you are unable to find an app that covers your needs, it is best to ask the online community if something exists before you begin development on your own.

CRM Website Themes:
With the addition of CRM and eCommerce modules in version 8.0, Odoo is working on building 70 themes to allow users to customize their publicly facing website.  These themes will be hosted on Odoo's theme store and during development their progress can be viewed on runbot.  These themes will not be free and Odoo encourages designers to develop themes that they can list and sell on the website.

Odoo has released some documentation on building your own themes and will also be authoring an official book as well.  More information can be found on odoo's forum and OpenDays presentation slides.

Testing:
Odoo provides a service to test build all of their proposed updates and versions before they are committed to the base, this service is called runbot and is available to anyone to login and test a feature.  All Odoo modules are installed on every build instance.  The administration Login and Password is: admin and admin.  There are no posted restrictions as to what you can do with these databases, however common courtesy should apply as you will not be the only person using them.  Also note that these instances can be eliminated at any moment.  The source code for runbot is available if you wish to host your own development environment internally.

Another service available for testing is demo.odoo.com which contains the demonstration data  that you can use to see the system in action.  It does not require a login or password.

Translations:
Odoo is available (to varying degrees of completeness) in about 80 languages and dialects.  The current master repository for all language translation work is being hosted on Launchpad for version 8.0 and below. Translations are broken up into three categories: Server, Web and Addons.  Starting with Odoo version 9.0, Odoo will find and use another platform such as "Transifex or [something] similar" according to the Odoo development wiki.

Problems / Support:
Depending on whom you wish to respond, you may ask for help on Odoo's forums or email lists (see the community section above), or you may engage with Odoo directly.  If you wish to report a bug or suggest an enhancement in the software, you may do so by posting a GitHub Issue.  You should search the issue database and comment on open issues relating to your problem and thus not open a duplicate issue for no reason.  For bugs you should verify that they still exist in the current development version by testing them on runbot (see testing above).  And then following the Contributing Guidelines and post your issue using the following format.  Please do not make extra work for the Odoo developers by posting duplicate issues that they must then merge or delete. 

If you have a question that is best directed to a support person (not a developer) then you may contact odoo support via the following email address: online@odoo.com  If you don't have a maintenance contract you can contact: sales@odoo.com

Upgrading:
See Cost above.

News / Keeping Current:
If you wish to know the latest news on Odoo development, releases, bugs and new R&D features there are several sources you should follow.  Odoo tweets on @odoo, @odooCMS, @odooStatus as well as posts on facebook. There are a few other resources including Linked-in, Google+, RSS feed etc that are mentioned by Fabrice Henrion (Director of Odoo - USA) in his forum post.

You can also "watch" the odoo and OCA development on GitHub to receive emails on new issues and merge requests.