The Xen (pronounced zen) hypervisor runs directly on the hardware and then starts a priviledged virtualized system called "Dom0". You can regard that as xen-controller. From the Dom0 system you can create other virtualised systems called Domains that Xen literature calls DomU.
Xen works with paravirtualisation and full virtualisation (full virtualisation is only available on systems supporting CPU virtualisation extensions). Paravirtualisation requires an OS that is aware of it to work but on the other hand provides almost direct access to critical parts of the hardware like memory and IO, making paravirtualisation extremely fast. Linux and most BSDs have paravirtualised awareness and are widely used with Xen.
Full virtualisation is used by operating systems that are not paravirtualisation-aware like Microsoft Windows and require hardware support, specifically the virtualisation extensions from AMD and Intel (AMD-V or VT-x respectively). Also, full virtualisation has to emulate most of the access to hardware, having a performance penalty to this kind of virtualisation.
Xen started as a project at the University of Cambridge in 2003 as an open source project with an associated company selling support services (XenSource, Inc). Citrix acquired XenSource in 2007 and renamed the professional services XenServer, while the open source project moved to http://www.xen.org and continued its development under the guidance of the Xen Project Advisory Board formed by companies interested in Xen development. XenServer products were renamed Xen Cloud Platform (XCP) and open sourced by Citrix in 2009 and are available for download on the xen.org website.