AM is an acronym for "Amplitude Modulation" and FM stands for "Frequency modulation". Radio waves carry communication signals by superimposing a sound wave on top of a "carrier wave" of a specific frequency. When the reciever is tuned, it filters everything except the carrier wave frequency and it's associated sound signal.
AM carrier waves have a set frequency. When the reciever "demodulates" the wave, it filters out the sound wave from the carrier. The sound wave is then amplified. The disadvantage of this system is that only 1 channel can be broadcast. AM radio was known as "mono" and didn't sound like a live musical preformance. It is still used for radio voice broadcasts.
FM radio carrier waves vary in frequency. This is demodulated into the second channel and true stereo could then be broadcast. This is nearly indistinguishable from a live music preformance.
AM and FM radios were analog devices. A radio wave of varying frequency and amplitude was changed directly into a sound wave. Broadband is entirely digital. Digital signals consist of a morse code pattern of identical radio pulses which is decoded by a digital reciever. Each set of signals represents the numerical equivalent of the amplitude of a sound wave. The digital reciever takes this data and uses it to reproduce its own set of sound waves. The advantages of digital transmission is that microwave frequencies can be used. The wavelength is extremly small, so more information can be broadcast. This is why this form of transmission is referred to as "broadband".
In terms of the number of channels that can be carried, AM has the lowest number of channels, FM is somewhat better and microwave signals can carry hundreds of separate channels.