A RC lipo battery balance connector is nothing more than a breakout connector for accessing all the individual cells contained in a lipo. This allows for the monitoring and manipulation of each cell individually. 

Why are balance connectors needed?
Like everything in the world, no two things are exactly the same, and batteries are no different. When multiple batteries, or cells, are assembled into packs, those different differences can lead to imbalances within the pack. In most cases, like NiXX cells, this is not a problem due to the chemistry of the cells. But when using lithium based cells, these imbalances can damage the cells and in some extreme cases even result in a fire. 

Looking at how RC lipo battery are wired
The best way to understand how something it put together is to take it apart. Well I don't have any lipos to take apart to show you but I can draw you a diagram to illustrate it.

As you can see this pack is composed of 3 cells wired in series. Then the main leads are connected to the outer most (-) and (+) tabs. The balance leads are connected at both the (-) and (+) of each cell. As such there will always be one more balance wire than the number of cells, in this case 4 wires for 3 cells. 

Monitoring and manipulating each cell
Finally it's time to get down to the real reason for having the balance connector, the ability to monitor and manipulate each individual cell. This ability exists because there is a wire going to the (-) and (+) of each individual call. As such you, or more often your charger, can use each pair to access a specific cell. For example below is the breakdown of cells (voltage) by color in the above photo when using a volt meter.

black -> yellow = cell 1 (3.7V)
yellow -> green = cell 2 (3.7V)
green -> red = cell 3 (3.7V)
black -> green = cells 1 + 2 (7.4V)
yellow -> red = cells 2 + 3 (7.4V)
black -> red = cells 1 + 2 + 3 (11.1V)

When a lipo's balance connector is plugged into a lipo balancer, the balancer can both monitor the voltage of each cell and if need be, discharge any high cells in order to balance the pack. 

Other uses for the balance connector
Some chargers only charge through the balance connector. Many cheap oem chargers do this but some nicer models will also allow for it. In this case the charger is both charging and balancing through the balance connector.

Sometimes there is a need for a connection to a battery and you don't want to use the main leads. In that case you could use the balance connector for a low current draw connection. For example lets say you have a camera setup on a bird. Instead of wiring the power for camera system into the main bird, you could connect a separate bec to the balance connector on the battery in order to power this separate system.

Charging one cell through the balance connector
Since the balance connector has access to each cell, it is possible to charge each cell individually. All that is needed is an adapter and a battery charger that can charge single cells.

In conclusion
As you can see balance connectors are not magical at all. They simply break out all the individual cells in a lipo so you or the charger can monitor and manipulate each individual cell.