C

mapnik
Mapnik is an OpenSource C++/Python toolkit for developing GIS (Geographic Information Systems) applications. At the core is a C++ shared library providing algorithms/patterns for spatial data access and visualization.
Source: http://developer.berlios.de/project/showfiles.php?group_id=3133

libral - Raster Algebra Library
libral is a C library for raster algebra, which is an algebra of two dimensional rasters, often associated with geoinformatics or geographical information systems (GIS). libral contains general raster algebra functions and functions for terrain analysis.

GeoServer
The Geoserver project acts as a thin, portable, OpenGIS-compliant web services layer on top of existing data sources.

SAGA GIS
SAGA -- System for an Automated Geo-scientific Analysis -- a free Geographical Information System (GIS)
The SAGA API supports grid data like digital terrain models andsatellite images, vector data, and tables.
This API makes it easy to implement new algorithms and exempts the developer from hassle programming overhead like user-interface design or file-io.
SAGA comes with a large collection of SAGA-Modules, in areas of terrain-analysis, geo-statistics, image processing and process simulation.

Quantum GIS
Quantum GIS (QGIS) is designed to be a Geographic Information System (GIS) built for Linux/Unix. QGIS will offer support for vector and raster formats.

Shapelib
Shapelib is a simple C API for reading and writing ESRI Shapefiles. It is Free Software available in source form, under an MIT Style (preferred) and GNU LGPL (optional) license