I am just starting out with virtualisation, in fact I don't use it for anything business critical because I don't know anywhere near enough about it!
I currently have two HP ProLiant DL360's with 4GB ram and two 3GHz processors each. They both have ESXi installed and running and I'm using a spare Windows XP machine to run vSphere client.
I have read into vCenter and wondered if there would be any benefit in using it in such a small setup? Also, would it require a completely dedicated server of its own?
I want to learn everything I can about vmware and virtualisation, but don't want to install something that won't benefit me in any way.
Yes, it is very helpful, even in small environments. One of the huge benefits you'll get by running you servers within VMware is that you're no longer tied to a specific piece of hardware. If your ESXi server croaks, you can easily move the files from the VMFS over to a new server and start them up without having to worry about changing drivers, incompatible HALs, etc.
Additionally, if you're like 99% of the businesses out there, your non-virtualized systems sit on hardware that is idle 99% of the time. By consolidating, you'll get better usage out of your hardware, possibly allowing you to decrease the number of physical systems needed.
Another benefit you'll get is the ability to create snapshots of your servers. This is very useful to do, say, before patching or other maintenance. If things to wrong, you can easily revert back to a known-good snapshot. Keep in mind that, for several reasons, it's not a good idea to keep large amounts of snapshots around for any one VM. So try and delete/consolidate snapshots when you've determined they're not needed any longer.
There are many other benefits and small conveniences you'll discover as time goes on. After getting used to operating in a virtualized environment, you'll find it pretty painful to go back to a non-virtualzed system.
To answer your other question directly:
If you have HA and have your VMFS volumes on shared storage, vCenter can run on one of the ESXi servers. If you are just using local disk and have no HA set up, then you'll want a separate server for vCenter.
I think there is some confusion here. I would HIGHLY recommend running the ESXi Hypervisor for the reasons that ErikA listed above (Machine Independence, Snapshots [although there are performance problems with having lots of snapshots and snapshots that stick around for a long time], and consolidation) and that is ultimately what ESXi gives you. The Hypervisor is what runs on the local servers to provide these.
VMware's vCenter gives you better management tools and if you purchase shared storage you get vMotion (move a VM from one host to another), High Availability (automatically reboot VMs if they or their host fails), and Fault Tolerance (run two copies of the same VM). vCenter runs on a separate Windows install (we're running it on a VM with the VM files stored on shared storage).
With as small as your servers are (4 GB of RAM won't hold a lot of VMs) I wouldn't recommend using vCenter to manage them. When you start looking at multiple VM hosts with 15+ VMs or shared storage then I would start looking at vCenter.
One of the things I like about VMWare (and virtualPC, and XEN) is the disaster recovery that was mentioned by ErikA. The other thing I like is that you can better isolate services. In the 'olden days' you would have a server sitting there, and when you needed something else managed by a server, you added it to whatever server had the least amount of load on it.. (or some other logical scheme) However, pretty soon, your file server is also hosting your calendar app, your backup software, some important printers, etc (that was my specific case once).. So, if I needed to install an update to the Backup Software because it had a big security bug, I had to reboot the server in the middle of the day, meaning people would get bumped out of their calendar app, not connect to printers, and not be able to access some of the file shares.. Running these all as VM's, you can Isolate them from each other, and re-assign. If you need to reboot the Backup Server, go for it, nothing else goes down. If the Calendar Serving tool becomes very popular and grows past the server capacity, you move it. (either offline at night, or a live migration) to a server that has less load. It allows you to be much more dynamic.