First of all, unless your family is already fairly tech-savvy, this doesn't sound like a very good idea to shuffle off onto them. Right now, virtualized 3D just isn't nearly as good as native. AFAIK, some light gaming can be done, but heavy gaming experiences are still sub-par. That's without getting into the whole problem of the actual usage experience -- keeping your family members on "their" VMs, etc. I just don't see this as being a very good idea or experience to be foisting onto other people.
All that said, if you're determined to do this, here's some of what you need to know.
Amongst your Type-2 hypervisors, VMware Workstation and Oracle's VirtualBox will both accelerate guest Direct3D and OpenGL with varying levels of success. I believe that Parallels Workstation product does as well. Any of these products are going to require a host OS running first, with the virtual machine started up afterward. All the benchmarks I've seen from these programs indicate that there's lots of compatibility issues, and performance takes a severe hit compared to natively using the same GPU.
If you decide to go with a Type-1 hypervisor, Microsoft's Hyper-V will probably be your best bet (at least with Windows guests), since it's RemoteFX technology will let you present virtual GPUs to Windows 7 guests. I haven't seen any benchmarks, but this blog post has a Crysis screenshot. I understand that KVM allows VGA Passthrough as well, but I understand that to be dedicating a GPU to a VM, which is probably not exactly what you're looking for. (Some Type-2 hypervisors can also do pass secondary graphics cards to VMs as well.)
The requirements are relatively straightforward, but finding parts (particularly from Intel) that support them can be... tricky.
From the CPU, you'll need Second Level Address Translation (SLAT) support. Intel calls it Extended Page Tables and AMD calls it Rapid Virtualization Indexing. Nested Page Tables is another name you might see. It was first seen with the Nehalem architecture (First Generation Core-i5/i7 Series) on Intel, and 3rd Generation AMD Opteron CPUs. According to AMD, pretty much any Athlon II, Phenom, or newer CPU should have it. Intel has a VT List on their ARK knowledge base. However, that only covers VT-x, not VT-d, and Intel uses VT-d support to segment their markets. (For example, the Sandy Bridge "K" CPUs, e.g. Core i5-2500K, do not support VT-d. Make sure your virtualization solution doesn't require VT-d before buying.)
The GPU needs to support DirectX 9.0c and 10.0. In other words, pretty much any modern AMD or nVidia GPU. For RemoteFX, despite Microsoft's anemic HCL, 3rd parties have indicated that pretty much any current GPU should work. This is a case where extra GPU memory is beneficial, so I'd be sure to go with at least 1 GB of RAM on the video card.
Beyond that, you're talking about a VM host. RAM and I/O are almost always issues under any kind of load. Thankfully RAM is relatively cheap -- there's no reason you can't have 16 GB of RAM on a relatively inexpensive desktop system. I/O is another matter -- if you're going to have multiple VMs running at once, getting them on their own hard drives is highly beneficial. SSDs are also an option, if you can afford the capacity you need.