What is the purpose of the capacitor in this current transformer to arduino interfacing circuit?
Most ADCs (analog-digital converters) are uni-polar. They accept voltages from 0 to some maximum; 0 to 5 V, for example.
The signal from the current transformer is AC and will alternate above and below 0 V. At best you won't be able to read the negative values. At worst the negative voltage will destroy your ADC.
To fix this you need to bias (offset) the received voltage to be centred at the mid-voltage of the ADC. This will allow positive readings and negative readings to be made. The bias must be removed in software before calculations are made.
R1 and R2 divide the supply voltage in two to give the mid-point.
C1 just gives some stability to the bias voltage to prevent it changing with power supply noise, etc.



  In the description of the schematic, in the section titled "adding a dc bias", the text states that capacitor C1 "provides a path for the alternating current to bypass the resistor."


A tiny AC current will flow into the ADC input. It has to return to the CT via the GND line. C1 will allow it to bypass R2.


  Is the text referring to the Burden resistor?


No. R2.


  Why is alternating current bypassing the resistor beneficial to this circuit?


Accuracy.


  The text also states that a value of 10uf is suitable for this capacitor.


The impedance of a capacitor is given by \$ Z = \frac {1}{2 \pi f C} \$. If you work this out for 10 uF and your mains frequency it should be very low in comparison with R2.


  Are there any other specifications of the capacitor that are important in the design of this circuit (such as polarity)?


With the 5 V supply and a 2:1 divider there will be 2.5 V across the capacitor. Top is positive. Polarity will matter for an electrolytic.