How does this single supply opamp circuit work, and what does the output look like?
It doesn't work.. at least it needs a power supply on the op-amp. 


The TL081 has a common-mode input range that goes down about 3V above the negative rail, so it cannot deal with voltages at the negative rail (in this case, ground). 
The reason to have the 10uF there is to have a DC gain of 1 and an AC gain that is higher (in this case the AC gain is 11 and it rolls off at the 16Hz -3dB cutoff frequency).
With the capacitor there, the DC gain is still one, so you would need to bias the input to +4.5V or so and capacitively couple the input to get it to work (say two resistors across the supply, with a capacitor to the input- but that has no power supply noise rejection).