Your first step is to download an app called Splashtop 2 HD, either from the App Store (iPhone version, iPad version) or from Google Play.
It can be a little pricey on iOS (especially when compared to free remote desktop apps like LogMeIn), but this one is specifically designed for games. That means you'll get low latency streaming - so no lag between inputs - and audio.
Once you've downloaded Splashtop 2 HD, find its icon and launch the app.
Now, it's time to open an account.
On the login page of the app, tap on the link that says 'New to Splashtop 2? Create an account'. You'll need to enter a valid email address and a password.
Over on your PC or Mac, you'll want to download and install the free Splashtop streamer app. We'll wait.
All done? Good. You'll see a status page (displayed above), so just log in to your account to get the streaming server up and running.
Back on your tablet, load up the app again and find the device list. This is a list of all computers running the Splashtop streamer that are logged in to the same account.
As long as both devices are on the same local network, you will see a black computer screen image (like above), and no satellite dish icon (which would indicate that you're on different networks).
You might want to tweak some settings before you start playing. If you tap on the 'edit' button, you'll bring up a list of resolutions to use. 1024x768 is the default and should be fine, but you can scale down to 800x600 if you're noticing any lag.
Also, make sure that you've got 'Optimize for your network' set to 'On' in the options (tap the gear icon on the device list).
Your options will change depending on the game. A twitchy action game needs better latency (so drop the resolution down and change to smooth mode), while a cerebral strategy game needs legible text (so put the resolution up and keep it on sharp).
You're good to go. Tap on the name of the computer you want to stream from, and you should see your computer's desktop appear on your iPad or Android tablet. Now, you can just use your computer like normal to load up your game of your choice.
Your mileage may vary from game to game. Half Life 2 loaded perfectly on first go, but Civilization V didn't show at first. To fix that problem, we had to load it in a small window on our Mac, then open up Splashtop and choose the 'Fullscreen' option in the settings.
You can use any controllers, or your mice and keyboard, to control the game, of course. Just make sure that they're plugged into your computer or within wireless earshot.
If you're on a PC, you can turn off your display by hitting the power button on your monitor. On Mac, it's not so straightforward. Open up the Splashtop streamer (right click on the icon in your menu bar, and choose 'preferences') and open the 'settings' tab.
Click 'Install driver' to install a virtual display driver. Once that's installed (and you've restarted your Mac), click 'Enable blank screen'.
Now, next time you log in with the app, your Mac will display a screensaver. Hoorah.