What kinds of inputs can an LDO take?
I just recently stumbled across LDOs and understand that they are used to regulate an output voltage. I understand that they require a differential amplifier, a reference voltage, 


This is essentially true for all linear regulators. An LDO regulator is a linear regulator with lower than usual minimum (Vin-Vout) value.
(Technically an "LDO" does not have to be a linear regulator BUT it tends to not make sense to use the term with most switching regulators. 

The minimum allowed (Vin-Vout)voltage is called the "dropout voltage".


  ... and a pass element of enormous size.


Pass element size need only be enormous if device rating is enormous.
As with ALL linear regulators the pass element dissipates
Pdis = (Vin - Vout) x Iout + small control power. 


  However, I'm unclear as to what inputs can be fed to an LDO. 


LDOs are usually monopolar - Vin and Vout are of the same polarity . You COULD make an LDO that output +/-Vreg (+ out when + in, - out when - in) when Vin was of suitably greater magnitude than Vout BUT there would be a "hole" in the middle when Vin is of lower magnitude than |Vout + Vdropout|


  For example, could I feed it a sinusoidal input and expect a dc output? 


For almost all applications with Vin is AC a rectifier is used to convert Vin_AC to Vin_DC. USUALLY this is filtered to produce DC of low-enough AC ripple.

If desired you could make Vin = +DC_offset + sine wave such that Vin is always >= Vin-min. Vout_DC will then be constant subject to the AC rejection specification of the LDO. If V_offset_DC + Vsine drops below Vdropout the output will decrease in a usually undefined manner. 


  Could I discharge a capacitor through an LDO and see a dc output at the LDO 


Presumably into a load.
As long as Vin >= Vout_specified+Vdropout then Vout will be constant. output? 


  Will an LDO be able to take any input and output a dc voltage?


As above - Vin must be >= Vout_spec+Vdropout.
Max Vin and Iin or Iout is specified in the data sheet.