The hardware requirements aren't steep. Just like running a game, the more processing power behind the host machine, the less taxing streaming will be on the system. But assuming you've got the hardware to accelerate the video encoding process (and almost every system sold in the last few years should), an adequate amount of memory, and decent broadband, then a fairly modest system will work just fine.
Both Nvidia's GameStream and Steam's In-House Streaming recommend using a quad-core Intel CPU (or six-core AMD processor). We'd also suggest having at least 8GB in the system. As always, more memory is better. The client systems have more modest requirements, however. They should have a GPU with built-in hardware accelerated H.264 decoding, but just about any modern computer or device should have that. If you can smoothly play high-quality HD video streams, you should be good.
The network requirements are a bit trickier and arguably more important. Not only must there be adequate bandwidth between the game streaming host and client system, but low latency is paramount as well. Ideally you'd have all systems wired to the same network via a network switch, but even a 100Mb connection would offer more than enough bandwidth. Nvidia recommends a 10Mb/s upstream and 2Mb/s downstream, which is nothing for a modern wired network.
Should you have to use a wireless network, a fast, dual-band 802.11ac or 802.11n router/access point would be the next best setup. Nvidia has even worked with a number of partners to certify wireless network devices as GameStream-ready.
Also note the clients connecting to the wireless access point should feature matching network controllers. It's best to be in close proximity to the router to ensure a strong signal, and to employ the cleanest channel to minimise interference. Our "Ultimate Router Guide" has stacks of suggestions. Streaming outside of your local network is possible, but you'll be at the mercy of your broadband provider.
Game settings matter – lower resolution will require less bandwidth, while turning off vertical sync will help minimise input lag. In-game image quality, resolution, frame rates, and bitrates can all usually be altered, and finding the right balance between image quality, bandwidth, and latency/lag is key to a good experience.
We've found that game streaming locally is typically awesome if you meet the minimum system requirements and have a fast home network. Steam's In-Home Streaming and Nvidia's GameStream on the Shield Portable and Shield Tablet often seem just as good as playing on a local PC.
Game streaming from the cloud isn't always perfect, but will improve as broadband speeds rise and more servers are introduced into the ecosystem. As Nvidia adds more servers, the user experience for GRID should improve since the load will be shared and gamers can access servers closer to their location.