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If FM radios work by modulating the frequency, how is it that we can tune into a specific channel, and hear a song or station? Wouldn't the channel need to be modulated along with the varying frequency?

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That is the work of Band-Pass Filter . en.wikipedia.org/wiki/Band-pass_filter –  ABC May 9 '13 at 2:56
The details of how FM works are a deeper question than just how signals can be selectively filtered, so I don't see this as a duplicate of 13599. –  dmckee May 9 '13 at 3:05

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The frequency of the signal is modulated in a relatively narrow band, and drives the audio circuit in proportion to the resonant power between the signal and a resonating circuit tuned to almost the range where the signal resides.

The result is that the power in the audio circuit varies with the frequency of the driving signal

Other signals are far from the resonant frequency of the tuner and do not drive it to an appreciable degree. Further regulatory steps are taken in most place to reduce insure a minimum difference between nearby stations.

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When you say other signals are far from the resonant frequency, do you mean that FM stations are fairly spaced out? –  user24082 May 9 '13 at 3:12
When you drive a resonator (within some limits in the power of the driver and the $Q$ of the resonator) the response is strongly dependent on the difference between the driving frequency and the natural frequency of the resonator divided by a width figure that is proportional to $1/Q$. The next station over differs in frequency by many times that width so there is negligible response from the resonator. –  dmckee May 9 '13 at 3:15
This is why FM radio stations in the US use the odd points only. 98.1, 98.3, 95.5, 98.7, 98.9, etc. (all in MHz). There are no .{0,2,4,6,8} FM stations. They are spaced out by 200 KHz even though each station is only about 25 - 50 KHz wide which gives about 75 KHz of padding on each side. –  Brandon Enright May 9 '13 at 23:48

Modulation, whether AM, PM, FM or whatever (even CW), necessarily widens the spectrum from that of the pure tone of the carrier.

Thus, in the design of any radio demodulator circuit, the bandwidth of the modulated carrier must be taken into account.

Generally, the RF signal is down-converted, via a mixer and local oscillator, to an Intermediate Frequency (or more modernly, the RF is directly converted to baseband) where a sequence of filters and amplifiers, with the appropriate bandwidth, attenuate frequencies outside of the desired bandwidth and amplify the frequencies within. This "groomed" IF signal is then passed to the demodulator.

For the commercial FM band, the channel spacing is 200kHz. But, the actual bandwidth is (from Wiki):

Bandwidth of 200 kHz is not needed to accommodate an audio signal — 20 kHz to 30 kHz is all that is necessary for a narrowband FM signal. The 200 kHz bandwidth allowed room for ±75 kHz signal deviation from the assigned frequency, plus guard bands to reduce or eliminate adjacent channel interference. The larger bandwidth allows for broadcasting a 15 kHz bandwidth audio signal plus a 38 kHz stereo "subcarrier"—a piggyback signal that rides on the main signal. Additional unused capacity is used by some broadcasters to transmit utility functions such as background music for public areas, GPS auxiliary signals, or financial market data.

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