I am no expert on this subject. I did not go to school to learn how to charge lipos nor did I even attend a 1-day seminar on the subject. Everything written below is from personal experience, online reading and basic understanding of electrical circuits. If you find a mistake, please let me know.
Lipo charging in basic terms
The goal of a lipo charger is to charge a lipo to a specified voltage. That voltage is based on the number of cells multiplied by 4.2V. The charger goes about this by adjusting the output voltage in order to make current flow to the lipo. Initially the charger raises the voltage enough to match the amount of current flowing to what the user has selected. This stage is referred to as the Constant Current, or CC phase. Current flows because of the difference in voltage between the charger and the pack. The greater the difference, the greater the charge current. During the charge process the resting voltage of the pack rises. Once the charge voltage reaches the maximum voltage of the pack, the charger holds it there. This stage is referred to as the Constant Voltage, or CV phase. The charging voltage is now held at the max pack voltage but current continues to flow because there is still a difference between the charge and pack resting voltages. As the pack resting voltage rises, the difference will become less and less until they are very close. During this phase the charge current will drop off more and more until it is very small. Once the charge current and voltage reach predetermined values, the charge cycle is complete.
Note on fast charging
One important concept that many people overlook when charging at rates beyond 1C is that every charge cycle is made up of both CC and CV phases. The higher charge rates do shorten the CC phase but can lengthen the CV phase. This means that the time savings is not linear to the charge rate. In other words a 2C charge cycle is not twice as fast as a 1C because the CV phase will last at least as long as the 1C charge cycle. It also means that as the charge rate is increased, the time savings diminish.
Lipo balancing in basic terms
Almost all new lipo chargers come with built-in cell balancers. Some older chargers have them but not many and so the use of an external balancer is required for balance charging. Like the name implies, this device is designed to balance lipos. But what does balance really mean? It simply means that it makes all the cells in a pack the same voltage. How does it do this? Well that too is fairly simple. The balancer uses the lipo balance connector to access each cell. Once connected it can do 2 things, monitor the voltage of each cell and if need be, discharge high cells in order to make all the cells the same voltage. That's it, it really is that simple.