GEOS is the software you get from http://trac.osgeo.org/geos/ (which has the title you mentioned, so it is likely what you meant).
There are C and C++ interfaces as part of a standard build. libgeos_c means the C interface. On Ubuntu (and probably Debian) it is the
libgeos-c1 Geometry engine for Geographic Information Systems - C Library
package. Other distributions will also likely have it as a package or build script. They may or may not split out the C++ interface.
I'd suggest using system packages (via your package manager) unless you have strong reason to prefer building from source. On Ubuntu, a command line install would be something like:
sudo apt-get install libgeos-c1
(there are many other options including GUI and aptitude client, but this should work for all installs).