Balanced demodulator
To clarify the terminology:


A modulator is a circuit that varies some property (frequency/phase, amplitude or some combination of thereof) of a signal (the carrier) according to another signal (the baseband).  
A demodulator reverses the action of the modulator, yielding the baseband from a modulated carrier.  Often a demodulator requires the unmodulated carrier as well, which is typically generated within a radio receiver by the local oscillator or LO.
A frequency mixer is a fundamental building block of modulators, demodulators and many other circuits. It combines two waveforms into the product of the two, as in it multiplies one signal by the other. This is useful, because such multiplication will shift every frequency component in one signal to sum and difference frequencies determined by the frequency content of the other signal.  

For example, mixing a 1 MHz carrier with a 10 kHz tone will produce a signal (specifically AM-DSSC) that contains both the sum of the frequencies of the two inputs (1.01 MHz) and the difference (0.99 MHz), but ideally no original 10 kHz or 1 MHz components. This operation is known as upconversion.

A frequency mixer can also be used as part of a demodulator. For example, mixing a modulated carrier that contains a 1.01 MHz and a 0.99 MHz tone with a 1 MHz tone will produce an output with two sum frequencies (2.01 MHz and 1.99 MHz) and, as both 1.01 MHz and 0.99 MHz map to the same difference frequency when mixed with 1 MHz, a single 10 kHz difference frequency. Low pass filtering the mixer output to remove the high frequency components would yield the original 10 kHz baseband signal. This operation is known as downconversion.


Given that both modulators and demodulators are often just mixers combined with filters, and that modulators and demodulators are in many cases be very similar circuits, the three terms are often (quite confusingly) used interchangeably. What people mean by balanced demodulator is really either demodulator which uses a balanced mixer or actually just (single or double) balanced mixer.  



What is a balanced mixer:

As briefly mentioned above, an ideal frequency mixer does not allow the frequency content found in the two inputs to pass unaltered to the mixed output, only the sum and difference frequencies of every input component. However, it may be desirable to allow significant frequency content from one or both inputs to pass directly trough to the output, as a tradeoff of otherwise better performance and/or lower cost.


A double balanced mixer attempts to block the frequency content of both inputs from passing directly to the output, only allowing the sum and difference frequencies trough. The classic ring modulator and Gilbert cell are examples of this type.  
A single balanced mixer allows frequency components from one of the two inputs to pass directly trough, but blocks the frequency content from the other input.  
An unbalanced mixer allows both to pass trough, so that the output contains both inputs a significant level in addition to the mixing products (and any harmonics etc.)  


The inputs bleeding trough to the outputs in an unbalanced mixer is caused by the mixer effectively adding a DC offset to the input signals. Balanced refers to the DC balance, or lack of a DC bias acting on the inputs, in a balanced mixer.
For example, plain amplitude modulation is done with effectively a single balanced mixer: The carrier can swing both positive or negative, but the baseband input is DC biased so that it never swings negative, thus never multiplying the carrier by a negative value. As a result, the carrier is still present at the output in addition to the sum and difference frequencies.