Between 2011 and 2012 Libre Office replaced Open Office.
The code for Open Office is now owned by the Apache Foundation but it is openly licenced. Open Office is likely to become a serious option again by 2013. Enhancements and bug fixes from both sources are freely available to both versions. This, and the spur of competition, is likely to lead to improvements in both.
Many former independent Open Office developers now support Libre Office. If you wish to install a free office system, Libre Office is the obvious current choice.
These pages provide a little background on why you might wish to make that choice.
For alternative online approaches, see Game Change?
TDF and Libre Office
The new independent Document Foundation (TDF) took much of the source code of OOo together with a couple of smaller code contributions from elsewhere and built an enhanced product, Libre Office (LibO). TDF is well placed organisationally and has attracted significant support.
The first production release of LibO 3.3.0 was on 25 January 2011. TDF maintain an aggressive release schedule, with two major releases each year plus minor releases roughly every month. Each major release is supported for about one year. Quality and functionality is improving rapidly. Generally the most recent release is recommended. To avoid security exposures and to benefit from improvements, installed version should be no more than 6 months old.
For document exchange, standardise on MS Office 97/2000/XP/2003 formats (.doc, .xls, .ppt etc). The variable and non-standard .docx, .xlsx, .pptx etc formats are best avoided. MS Office users can set the applications to automatically save in 97/2000/XP/2003 formats.
The ODF (Open Document Format: .odt, .ods, .odp etc) will become an increasingly viable option as governments, businesses and MS Office increasingly adopt it as their standard format.