The OGR Simple Features Library is a C++ open source library (and commandline tools) providing read (and sometimes write) access to a variety of vector file formats including ESRI Shapefiles, S-57, SDTS, PostGIS, Oracle Spatial, and Mapinfo mid/mif and TAB formats.
OGR is a part of the GDAL library.
ogr_api.h: OGR C API
The best way to get OGR utilities in ready-to-use form is to download the latest FWTools kit for your platform. While large, these include builds of the OGR utilities with lots of optional components built-in. Once downloaded follow the included instructions to setup your path and other environment variables correctly, and then you can use the various OGR utilities from the command line. The kits also include OpenEV, a viewer that will display OGR supported vector files.
The source code for this effort is intended to be available as OpenSource using an X Consortium style license. The OGR library is currently a loosely coupled subcomponent of the GDAL library, so you get all of GDAL for the "price" of OGR. See the GDAL Download and Building pages for details on getting the source and building it.
A gdal-announce mailing list subscription is a low volume way of keeping track of major developments with the GDAL/OGR project.
The firstname.lastname@example.org mailing list can be used for discussion of development and user issues related to OGR and related technologies. Subscriptions can be done, and archives reviewed on the web.
In addition to the C++ API primarily addressed in the online documentation, there is also a slightly less complete C API implemented on top of the C++ API, and access available from Python.
The C API is primarily intended to provide a less fragile API since slight changes in the C++ API (such as const correctness changes) can cause changes in method and class signatures that prevent use of new DLLs with older clients. The C API is also generally easy to call from other languages which allow call out to DLLs functions, such as Visual Basic, or Delphi. The API can be explored in the ogr_api.h include file. The gdal/ogr/ogr_capi_test.c is a small sample program demonstrating use of the C API.
The Python API isn't really well documented at this time, but parallels the C/C++ APIs. The interface classes can be browsed in the pymod/ogr.py (simple features) and pymod/osr.py (coordinate systems) python modules. The pymod/samples/assemblepoly.py sample script is one demonstration of using the python API.