The Yagi is one of a class of antennas known as parasitic antennas. They are called so because as baffo states, only the driven element of the yagi is actually directly driven by an RF energy source.

What makes these antennas work is what is called mutual coupling between the antenna elements. The electric field produced by the driven element exictes electric currents on the other elements, such as the reflector and the directors. This coupling is what leads to the connotation of "parasitic".

If the element lengths are chosen properly, the elements act as an array where there is constructive interference in the direction of the director(s), and destructive interference in the reverse direction. Multiple directors can be employed to increase the effective gain of the antenna. Yagi antennas with 15 to 20 directors are not uncommon at VHF frequencies.

Electrically, Yagis tend to have driving point impedances in the range of 10-30 ohms, with some capacitive reactance due to the shortened length of the driven elements. They cannot be mated directly to commonly used 50 ohm coaxial cable. A matching section is often employed to obtain a close match, such as the gamma match, the tee match, and my favorite, the hairpin match.