Hyper-V core is free, but to get what you want you need to purchase System Centre Virtual Machine Manager to do it.
VMWare ESXi is free, but to get what you want you need to purchase vSphere.
Xen is free, but the free version is always one generation behind the paid I believe.
To do what you want with VMWare and ESXi you'll also need shared storage. To do this with Xen you can use DRDB to replicate, but we've done that in the past and performance can suffer vastly.
As DanBig said, as far as the term "snapshots" goes, in the virtual world a snapshot is linked to the VM and if seperated is totally useless, so they're no good for backups. But all of the solutions have their own backup procedures.
To go into more detail, a "snapshot" is simply a point-in-time frozen version of the HDD, where (in layments terms) all changed bytes are linked to seperate sectors, so that both versions can co-exist. If you seperate one from the other, they're useless on their own, they both need to be kept together to be useful. So you can't just copy a snapshot and expect it to work as a backup. However, all hypervisors have their own backup techniques.
A fairly global one is cloning the VM, as they CAN be seperated from their original (unless you create a "linked" clone), but cloning is an IO-expensive operating, whereas snapshots require almost no IO do. Also, depending on the guest OS and if it supports the various hypervisor "tools" you may not be able to clone a machine while it's booted up (Hyper-V is a good example of this).