A Balun is used to "balance" unbalanced systems - i.e. those where power flows from an unbalanced line to a balanced line (hence, balun derives from balance to unbalanced). As an example, consider a coaxial cable connected to a half-wave dipole antenna shown in Figure 1.
Figure 1. An unbalanced transmission line (coaxial cable) connected to a dipole antenna.
In Figure 1, a coaxial cable is connected to a dipole antenna. For a dipole antenna to operate properly, the currents on both arms of the dipole should be equal in magnitude. When a coaxial cable is connected directly to a dipole antenna however, the currents will not neccessarily be equal. To see this, note that the current along a transmission line should be of equal magnitude on the inner and outer conductors, as is typically the case. Observe what happens when the coax is connected to the dipole. The current on the center conductor (the red/pink center core of the coax, labeled IA) has no where else to go, so must flow along the dipole arm that is connected to it. However, the current that travels along the inner side of the outer conductor (IB) has two options: it can travel down the dipole antenna, or down the reverse (outer) side of the outer conductor of the coaxial cable (labeled IC in Figure 1).
Ideally, the current IC should be zero. In that case, the current along the dipole arm connected to the outer conductor of the coax will be equal to the current on the other dipole arm - a desirable antenna characteristic. Because the dipole wants equal or balanced currents along its arms, it is the balanced section. The coaxial cable does not necesarily give this however - some of the current may travel down the outside of the outer coax, leading to unbalanced operation - this is the unbalanced section.
The solution to this problem, however you come up with it, is a balun. A balun forces an unbalanced transmission line to properly feed a balanced component. In Figure 1, this would be done by forcing IC to be zero somehow - this is often called choking the current or a current choke.
There are many baluns that have been developed to choke off the outer current and restore balanced operation. Some of the most popular methods are described in the following pages.
Baluns are pretty tricky to fully understand and at first pass can be difficult to grasp. However, suffice it to remember that a balun forces unbalanced lines to produce balanced operation, despite their inherent asymmetry.