glibc is the GNU C library. In a Unix system, the C library, often called just libc, is the main programming API to user programs. This library contains everything from the printf() function to sorting, networking, what-ever, functions.
It's currently maintained by the GNU people, and its current version is 2.2, but in Linux systems is installed with the version 6 (libc6) because before Linux had its own version of the GNU C library 1.0.