Servos are the little motors that actually drive the control surfaces on the plane up and down. A normal trainer plane will require four servos, and therfore a four-channel radio, since you must be able to control each servo independantly. Some trainers advertize only requiring three channels, but these can be awkward to learn on. You can always buy a radio with more channels than you need and just not use them. If you have aspirations to fly that P-40 Warhawk with retracting landing gear than you will need another channel to control the gear.
Radios are available from single channel up to ten channels - priced accordingly.
These are not legal in the USA for use on aircraft. Radios come in 'Dry Cell' or 'Ni-Cad'. This refers to the type of battery. Dry Cell batteries must be replaced when drained. Most Ni-Cads will last approximately 500 charges and cost about 4 times that of Dry Cells.