There are standards, but the topic is a bit...complicated. Details vary by chemistry, but for example with standard Li-ion batteries, charging is normally done at a "C-rate" of 0.5C - 1.0C to an end-of-charge voltage of 4.2V, where C is the battery capacity. Thus, a 3,000 mAh battery would be charged at 1.5 - 3.0 Amps. Note that charging at 1C means that the battery will be charged theoretically in one hour*, whereas 0.5C will take twice as long. The C-rate can be pushed to 2C or even 3C, but this leads to more rapid degradation of the battery.
So, for Li-ion, I suppose anything above 1C would be considered "fast charging". To do this without prematurely wearing out the cells, more sophisticated approaches to voltage and temperature regulation are needed, which requires different charger and battery designs. This is what Qualcomm is doing with their "Quick Charge" technology.
A basic overview of these approaches is provided
here.
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*This is not true in practice because normally when the cell reaches the final voltage (4.2V), the charger begins to decrease current in order to hold voltage at the target value. Charging then proceeds until the current drops to a specified level--I usually terminate at 0.1 or 0.2 A when charging my batteries. This slows down the final phase of charging significantly.