A transmitter, or 'Tx', is a plastic shell with control sticks and various switches, internal control logic, and perhaps an LCD. It also has a self-contained radio module that plugs into the back, which takes the serial communication stream from the transmitter and generates a wireless communications signal, which it outputs to the antenna that is plugged in. The signal goes through space and is picked up by a receiver, or 'Rx'.
There are two styles of transmitter: the twin-stick aircraft style, and the pistol-grip ground style.
Usually, the antenna shipped with a transmitter is a whip-style monopole or plastic-encased dipole antenna.
Radio bands are associated with broad regions of the electromagnetic spectrum. Usually, a radio module is designed to work on one band only. Radio channels are small sections of a radio band on which coherent signals are broadcasted by a unit. Radio control was traditionally accomplished using standardized 'narrow-band' (single channel) signals, but the advent of 2.4Ghz operation brought in various flavors of spread spectrum frequency-hopping capability, making interference between transmitters a lot less of a problem.
The physical response of the control sticks varies: the throttle axis is left to float at whatever position it's left at, while the pitch, yaw, and roll axes have springs attached which re-centers them in the event that the user's hands come off the controls.
The configuration of these axes varies as well: there are different ways of selecting which axis of which control stick is mapped to which control surface. With the constraint that throttle and pitch are always on vertical axes, that leaves four different possible transmitter modes, of which two are popular: Mode 1 in the US, and Mode 2 in the rest of the world.