To minimise the risk of neck strain, you should position the screen straight on so you are look directly at it. The top of the screen should be in line with your eyes. This way you don't need to look down at whatever's onscreen, and you can hold your head in a more natural position, avoiding neck strain and staving off headaches.
Some screens are wall-mountable, so this could be another way of ensuring the screen is positioned at the right height for you. Another alternative is a stand that raises up the screen from the desk.
The resolution, viewable screen size and response rate will be of most importance for entertainment and gaming duties. Choose a screen with no more than an 8ms (preferably 5ms) response time. A 22in screen with a 1920x1080-pixel native resolution is likely to be better value than a 24in model with the same resolution. A pixel pitch of 0.25mm or 0.26mm is ideal. On a 19in to 21in display, a 1366x768 pixel resolution will suffice.
If more than one person is going to be looking at this screen on a regular basis - perhaps the screen is an information display, for example - the viewing angles are very important. You want at least a 170-degree viewing angle vertically and horizontally. The horizontal figure is more critical.
Brightness and contrast govern how distinct things look onscreen. Choose a 1000:1 true contrast ratio and a brightness rating of at least 300 candelas per square metre.
HDMI and either DVI or DisplayPort connectors are the best for home entertainment and gaming. You may need a VGA connection if you're connecting two screens to your computer. DVI is a must.
Height adjustment is desirable but VESA wall mounts and monitor stands could be a good alternative.
All modern displays should have a label proving their EnergyStar-compliance. A three-year warranty is standard for displays.