You must have seen the radio that your grandfather loved listening to as you grew up.  I bet you loved turning the tuner knobs either way when no one was around!

You always wanted to know what the letters “AM” and “FM” meant. Today lets try to decode one of them. But before that, lets try and understand – What is modulation? Modulation is the process of changing a characteristic of a carrier wave with respect to a modulating wave. This modulating wave is otherwise known as the message signal which has the information to be transmitted. For example in amplitude modulation (AM), the amplitude of the carrier signal is varied in accordance with the modulating signal.
Amplitude modulation, more popularly known in its abbreviated form as “AM”, is a method of communication used mostly in the form of radio waves.  Amplitude modulation was originally engineered for use with the electric telephone (modern-day telephone) to add audio to a receiver connected to a telephone transmitter. In 1906 conducted an audio exhibition using amplitude modulation to broadcast audio signals using radio waves. From the exhibition in 1906 to modern day radio broadcasting, AM is still being used, referred to as the “AM Band”.

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Another way is FM (Frequency Modulation). While FM offers greater clarity for audio, and the higher frequencies that FM use offer a wider bandwidth, allowing for more information to be transmitted, one application where FM and digital are not suitable are Aviation communication, which to this day still use AM analogue. This is because weaker signals can be heard over stronger, closer ones with AM, allowing for emergency transmissions to have more chance of being heard over other traffic. Also, AM uses a narrower bandwidth than FM, allowing more users in a smaller space. This is important for the lower frequencies of Radio, where space is at a premium.
The next question is why do we have to modulate any signal? The alternative to modulating a wave and transmitting it is to send the wave as it is un-modulated. The first obstacle in doing that is the receiving the information. For efficient radiation and reception the transmitting and receiving antennas would have to have lengths comparable to a quarter-wavelength of the frequency used. In other words the shorter the wavelength (higher the frequency) the shorter the antenna need be. As a result of modulation, the low frequency message signal is ‘carried’ by a wave with a higher frequency.
The transmitted messages in AM radio are mainly music and voice in the audible range of 20 – 20 kHz. Due to this all the signals from all the AM stations will be mixed up in a very small band of frequencies. It would be difficult to separate the required message signals at the receiver unless they are converted to different frequencies of the electromagnetic spectrum. At the receiver a tuned circuit will be present to extract the required frequency for demodulation.
Let us see a circuit in which the given carrier is modulated depending on the modulating signal.
AM Modulator Circuit
The BJT must be biased properly to get a correct output. The carrier is connected to the base and this is amplified by the BJT and flows in the collector, but again the current is restricted by the opposing flow from the modulating wave connected to the emitter. As a result the output amplitude is varied according to the opposing modulating voltage.
One thing should be maintained in the AM circuit. That is that the message signal voltage (Vm) should be less than the carrier voltage (Vc). This relationship is known as the modulation index (Vm/Vc) and it should be between 0 and 1. When it is more than 1, or if the Vm is more than Vc, then distortion occurs. Let’s see the output of the above circuit – the modulated wave. 
AM Modulated Wave
Verify the frequency of the signal to be equal to the carrier frequency. Also note that the envelope of the modulating wave varies with reference to the carrier wave. So the amplitude of the modulating wave should be less than the carrier signal amplitude else the wave distorts. If the wave is distorted then a portion of the trough in the modulating wave will be clipped. As a result information is lost. You can see it practically by varying the amplitudes in our simulation and see the effects.
The expression for the modulation index is given as m= Vm/V= (Vmax – Vmin)/(Vmax + Vmin). Vm = (Vmax – Vmin)/2 and Vc = (Vmax + Vmin)/2. Refer the diagram given below.
AM Wave Voltages
The frequency spectrum would give the frequency components in the AM modulated wave. The frequency spectrum cannot be plotted like the frequency response. But you can reads more information about it from here:
Some useful video links are this and this.