Screens are measured diagonally, usually in inches. The most common sizes at the moment vary between 19" and 24".
There are two shapes of screen: normal (4:3 ratio) and widescreen (16:9 ratio). Widescreen is the current standard.
If the height of the screen is important (e.g. you need to see a whole page of text or to see as much of a spreadsheet as possible), then you might be better choosing a standard (4:3 ratio) screen.
Connecting to your screen - there are four common methods of connection.
VGA - the most common - a 15-pin analogue connection
DVI - these connections can be analogue (DVI-I) or digital (DVI-D)
HDMI - these digital connections are standard on most laptops or TV/entertainment equipment
mini-HDMI - these smaller connections are often found on small laptops and tablets
Connectors are available to convert analogue VGA to DVI-I, and digital DVI-D to HDMI.
DisplayPort (and Mini-DisplayPort) - this is the newest type of video connection type.
(Apple calls the Mini-DisplayPort "Thunderbolt".)
Connectors to convert to other types of video output are readily available.