The Hurd is firstly a collection of protocols formalizing how different components may interact. The protocols are designed to reduce the mutual trust requirements of the actors thereby permitting a more extensible system. These include interface definitions to manipulate files and directories and to resolve path names. This allows any process to implement a file system. The only requirement is that it have access to its backing store and that the principal that started it own the file system node to which it connects.
The Hurd is also a set of servers that implement these protocols. They include file systems, network protocols and authentication. The servers run on top of the Mach microkernel and use Mach's IPC mechanism to transfer information.
The Hurd provides a compatibility layer such that compiling higher level programs is essentially transparent; that is, by means of the glibc, it provides the same standard interfaces known from other UNIX-like systems. Thus, for a typical user, the Hurd is intended to silently work in the background providing the services and infrastructure which the microkernel itself has no business implementing, but that are required for higher level programs and libraries to operate.
The Hurd supplies the last major software component needed for a complete GNU operating system as originally conceived by Richard M. Stallman (RMS) in 1983. The GNU vision directly drove the creation and has guided the evolution of the Free Software Foundation, the organization that is the home of the GNU project.
The Hurd development effort is a somewhat separate project from the Debian GNU/Hurd port.
Read about what the GNU Hurd is gramatically speaking.
Read about the origin of the name.