Can anyone provide a brief explanation in layman's terms (as in "management friendly") of just what vSphere is, what can be done with it and why it's better than the alternatives?
Also, examples of possible real-world applications appreciated.
vSphere allows you to run multiple virtual machines on a single piece of hardware. Basically you take a single machine with say 16 CPU cores and 64 Gigs of RAM, and you slice that up into smaller bits so that you can run a bunch of smaller machines on it. Say 10 different virtual machines with 2 CPUs and 4 Gigs of RAM each. Some of these virtual machines can run Windows, some Linux, some Unix, etc all on a single piece of hardware saving you money on hardware, power, cooling, rack space, etc.
The biggest competitor to vSphere is Microsoft's Hyper-V. vSphere is a more mature product which gives it a leg up. The biggest benefit that vSphere has over Hyper-V are:
Now do keep in mind that not every server can be virtualized. Some servers just aren't good candidates for making them a VM, however that said, in most companies there is no reason that most physical servers couldn't be virtualized.
MRDenny's answer is extremely misleading as "vSphere allows you to run multiple virtual machines on a single piece of hardware" is a definition for any VM technology (QEMU, KVM, Xen, VirtualBox, VMware player/server/...).
vSphere has a small component that allows running VMs (basically equivalent to QEMU or VMware player), but it is almost entirely a set of tools to configure and manage virtual machines and virtual machine instances. This is done via a set of web GUIs connecting to various service daemons.
Its value is not in the ability to run multiple VMs on a single server (again, any VM technology can do that), but in making the creation and maintenance of those VMs easier.