Before Doing Anything Else

Ensure that all devices are powered on and connected. Make sure all cables are properly seated in their spots!
Follow the chain of connection starting with your laptop/device, and ending with the TV/Projector.
Restart any "Smart" Devices (Your laptop, Smart Projectors etc)

The TV/Projector Won't Turn On

Try using the buttons on the TV/Projector, it may just be batteries in the remote (Stationary Cupboard at the Office has batteries)
Check that it's still plugged in and the switch is turned on. This may also require checking the RCD is on and active (look for all plugs nearby with blue/green button and ensure the little light is on by pressing "reset" if not)
If it's still not working, contact the IT Department.

The first step in any troubleshooting process is to ensure that all devices are powered on and connected. Make sure all cables are properly seated in their spots!
Identify the cable you have plugged into your laptop, if any (the section linked here should help)
Find the Source Select button on the remote or screen you're attempting to connect to and press it until it matches your input. (Sometimes labelled "Input" or a little box with an arrow, often at the top right of the remote)
Make sure you've instructed your device to output to the projector or TV too! (Here's How)
If you're still having issues, contact the IT Department.

Check all cables are connected properly.
Look for the Focus controls on the projector. This is often a slide or a ring around the lens itself.
Play with settings until you get it right!
If it's still not working, contact the IT Department.

Check all cables are connected properly.
Make sure your resolution on your laptop/device is correctly set to the external display:
Right click on your Desktop (Windows) and choose "Display Settings"
Click on the display you have plugged in (If you're not sure, click "Identify")
Look for "Resolution" and change that to the recommended setting.
Play with settings until you get it right!
If it's still not working, contact the IT Department.

If you're using a VGA cable, check all cables are connected properly, as this is an RGB format, and one part of the cable or connector is almost ALWAYS at fault.
Sometimes the projector/TV/Monitor is set to a different colour temperature or setting, which will need to be changed. Look for settings/menu, and colour temperature.
If it's still not working, contact the IT Department.