What do inches mean on sockets, wrenches, etc.?

What do inches mean on sockets, wrenches, etc.?  
Which is bigger, 1/4 or 1/2?
1/4 is bigger than 1/6 and 1/2 is bigger than 1/4

1" (inch) = 25.4mm
The following table shows the size in inches and its equivalent in millimeters.
 
Diameter Diameter
inch mm
1/6 4,23
1/4 6,35
1/2 12,7