Why is the time constant 63.2% and not 50% or 70%?
Other answers haven't yet hit upon what makes e special: defining the time constant as the time required for something to drop by a factor of e means that at any moment of time, the rate of change will be such that--if that rate were continued--the time required to decay to nothing would be one time constant.

For example, if one has a 1uF cap and a 1M resistor, the time constant will be one second.  If the capacitor is charged to 10 volts, the voltage will fall at a rate of 10 volts/second.  If it's charged to 5 volts, the voltage will fall at a rate of 5 volts/second.  The fact that the rate of change decreases as the voltage does means that the voltage won't actually decay to nothing in one second, but the rate of decrease at any moment in time will be the current voltage divided by the time constant.

If the time constant were defined as any other unit (e.g. half-life), then the rate of decay would no longer correspond so nicely with the time constant.