Choosing the right type of wire?
The maximum current a wire can carry depends on several factor:  


wire diameter, obviously (Kortuk may take exception on me using the word obvious again, but this time it really is; anyone older than 10 will understand that thinner wires will allow less current through them)  
wire material. Often copper, but in some applications where weight is paramount aluminium may be used. Think overhead high-voltage power lines.  
insulation. The energy dissipated in the wire will cause a temperature rise, but the heat will also be given to the environment. The type of insulation determines the amount of thermal isolation. It also determines the maximum temperature allowable. Think melting of the insulation. edit: insulation will also determine maximum voltage. The insulation may breakdown when too thin in combination with a high dielectrical constant.  


This document gives the maximum current for copper wire as a function of diameter and temperature rise in free air at 30°C ambient temperature.