It's pretty straightforward. A balanced connection has three distinct and separate conductors; one signal negative, one signal positive (normally referred to as "hot") and one that is used for chassis ground and/or signal shield.
In contrast to an unbalanced connection where the signal negative and the chassis ground are both sent on the same conductor. In some cases the chassis ground connection is unused, in others the signal shield and chassis ground and the signal negative all reside on the same connector. Regardless, you have two connectors and three potential "users".
Separating the signal negative and the shield/ground connection offers a number of advantages. Although it will increase cable capacitance, you can have a shield connection that protects the signal carrying wires from magnetic and radio frequency interference. It's relatively easy to insure the capacitance effect is equal on both the signal negative and signal positive connections.
Impedance matching is often easier with a balanced configuration, and in many cases the signal operates at the "Professional" 600 ohms standard. This allows for very long cable runs without significant signal degradation in comparison to an unbalanced cable. Even with conventional Home Audio impedances, longer cable runs are possible. Broadly speaking a balanced line of 100 feet is capable of the same signal quality as an unbalanced line of 10 feet in length.
It's possible to reduce noise with a balanced connection, as some noise will cancel out (any noise that is present on both the negative and positive signal wires and in opposite polarity), depending on exactly the nature of the noise and the configuration of the cable. A more robust shielding opportunity also exists, which may prevent noise from entering the system in the first place.
Downside is increased cost of cable, in some cases connectors, and added complexity in the equipment itself. Some method must be used to accommodate the balanced configuration. It's possible to use what is called a Pseudo-Balanced connection, where you simply make a cable that adapts the balanced cable to an unbalanced input on the electronics.
Alternately you may preserve the true balanced configuration; this may be relatively simple (using a THAT 1200 circuit interface; THAT is a company created by former DBX engineers and they offer an IC to interface balanced inputs and outputs), it can be done by transformers (which increases cost) or the entire circuit may be balanced with separate electronics operating on the negative and positive legs of the circuit (now we are essentially doubling the cost of the circuit).
