Xen is a secure baremetal hypervisor (xen.gz), around 2 MB in size, and it's the first thing that boots on your computer from GRUB. After Xen hypervisor has started it boots the "management console" VM, called "Xen dom0", which is most often Linux, but it could also be BSD or Solaris. Upstream Linux kernel v3.0 can run as Xen dom0 without additional patches. Xen dom0 has some special privileges, like direct access to hardware, so you can run device drivers in dom0 (=use native Linux kernel device drivers for disk/net etc), and dom0 then provides virtual networks and virtual disks for other VMs through Xen hypervisor. Xen also has the concept of "driver domains", where you can dedicate a piece of hardware to some VM (with Xen PCI passthru), and run the driver for the hardware in the VM, instead of dom0, adding further separation and security to the system. Xen "Driver domain" VMs can provide virtual network and virtual disk backends for other VMs.
KVM on the other hand is a loadable module for Linux kernel, which turns Linux kernel into a hypervisor. The difference is that in KVM all the processes (sshd, apache, etc) running on the host Linux and the VMs share the same memory address space. So KVM has less separation between the host and the VMs, by design. VMs in KVM are processes on the host Linux, not "true" separated VMs.