What parameters affect battery charge time?
This is a very complex question that needs better definition to be answered well.

A major factor is the battery chemistry. Possble chemistries are LA (lead aid), Lithium Ion (LiIon), LiPo (lithium Polymer) , NiCd, NimH, LiFePO4.
 There are others but those are the most common.

Charging voltage is usually related to charging current. In some chemistries there is an upper limit to what can be used. You may be able to exceed this only once. eg LiIon usually charges at a max of C or C/2 in some cases.  

NimH can be charged at C rate (eg 2500 mAh cells at 2500 mA with available standard fast chargers. Faster rates (2C, 4C) are available for some cells with special chargers and maybe with internal monitoring of cells. 

LiPo (Lithium Polymer ) varies by manufacturer and product but rates of 10C and more are known.

A more specific question may get more details.