I know of the transponders used for communication by Air Traffic Control, but apart from that, what radio equipment do pilots use to talk to ATC? What modulation method is used?
As of 2012, most countries divide the upper 19 MHz into 760 channels for amplitude modulation voice transmissions, on frequencies from 118–136.975 MHz, in steps of 25 kHz.
A typical plane often has two communication radios, referred to as COM1 and COM2.
Each radio has an active and standby frequency. The pilot can listen to either radio, or both radios at the same time, but can only broadcast on a single radio.
There are often 2 navigation radios as well, NAV1 and NAV2, which can be tuned to VORs or other radio beacons, and the pilot can listen on those radios (typically for Morse Code IDs, Recorded Voice broadcasts, and occasionally, Voice-over-VOR ATC)
The major communication tools are obviously Vhf radios for voice, another one is HF, particularly in remote place or oceanic. In tran atlantic the voice comms are thru HF with SELCAL( selective call) which brings another communication tool more and more used : CPDLC( controller pilot datalink communications) this one is not by voice. Clearances or requests are given by this method certainly for transatlantic and more and more in land as well.
A short summary, specific to the US:
VHF AM is the most common voice mode for communications with ATC.
UHF AM is also used for military and some government services, but generally just military, for communication with ATC.
HF radio is used for communications with ATC, and is most often used on oceanic routes. It is not as "reliable" as VHF, but the signals will propagate farther.
Other enhancements and other modalities are available but they are not strictly voice communications.
14 CFR 87 covers aviation services, and defines frequencies and modes as well as uses for aviation specific services.
This is a very broad question just because every type of aircraft has different requirements for its design purpose. As civil aircraft may not need to use UHF frequencies where some military aircraft must have an UHF radio. And transponders are not used for communication. :)