Why does the diameter of household wire affect the maximum safe current? The greater the diameter of the wire used in household wiring, the greater the maximum current that can safely be carried by the wire. Why is this? Does the maximum safe current also depend on the length of the wire? Does it depend on what the wire is made of? 
 A: A larger diameter wire has lower resistance.
A lower resistance wire produces less heat for a given current flowing through it. 
The less heat the wire produces, the lower its temperature will be while operating.
Therefore a larger-diameter wire will be able to carry a larger current before it heats up enough to melt or scorch its insulation or ignite other objects near it.

Does it depend on what the wires made of?

Anything that lowers the resistance of the wire will improve its current carrying capability (aka ampacity). So using a material with lower resistivity will also help. 
However it might be cheaper to use a slightly higher resistivity material like aluminum than a lower resistivity material like copper, but simply make the aluminum wire thicker. Unfortunately, aluminum wire has other issues (work hardening and oxidation) that caused problems when we tried to use it for residential wiring in place of copper.
A: Wire is designed to be safe, if it gets too long appliances may not work so well but the wire won't overheat because it's so long.  The width was designed to be safe for all lengths but wire can still overheat if you put a tight bend in it ( so it's important your electrician does a good job and follows code) and it can overheat if you put too big a fuse and run too many appliances on the same wire.
