Why doesn't the motion of a car affect the frequency of radio stations? When we go in a car and tune to an FM radio station, why doesn't our motion disturb the frequency?
Like the Doppler effect?
 A: To further add to John Rennie's answer: you don't even need autotuning for a frequency drift of the magnitude John calculates (10Hz): all FM receivers I've ever dealt with (I qualified as an electrical engineer in 1985 and worked a few short years in communications before returning to study) demodulated with a phase locked loop detector, whose job it is to follow drifts in the carrier frequency like this.
In any case, in any FM demodulation, a constant 10Hz offset decodes to a very small DC (*i.e.zero hertz constant offset) signal: let's say the full dynamic range corresponds to $10^5Hz$ as in John's answer, this offset is 80dB down on the demodulator's full dynamic range. The fully demodulated signal probably doesn't reproduce signals down to DC anyway. Your car's speed cannot change quickly, so variations in its speed would correspond to a variation in this offset with a frequency of a hertz or so maximum: if it gets through the full demodulation filter at all, it's going to have negligible effect on the signal.
A: John and Rod already pointed out that the expected frequency shift from driving a car is "small"; I would like to expand a little more on the way FM works.
FM = Frequency Modulation. The carrier (nominal center frequency) is being modulated - that is, in order to convey the audio content, the frequency is actually moving around deliberately in order to get the music into your car. And it moves quite a lot.
In the case of FM, you have to worry about two things in the modulation: 


*

*What is the relationship between amplitude range you want to convey, and the frequency shift?

*What is the highest frequency you want to transmit? 


Some of the following facts were taken from this presentation.
The frequency response is typically 50 Hz to 15 kHz; 100% modulation is defined as 75 kHz on either side (max amplitude). For stereo FM, the highest modulating frequency is 53 kHz.  According to Carson's Rule, the minimum channel width should be $CBR = 2(\Delta f + f_m)$ where $CBR$ is the bandwidth requirement, $\Delta f$ is the frequency swing corresponding to full amplitude, and $f_m$ is the maximum modulation frequency (53 kHz). This means that an FM transmitter needs 256 kHz according to this rule - reason why geographically close transmitters are never next to each other on the dial (channels are allocated on 200 kHz slots, but you can't transmit at 90.1 and 90.3 in the same town and get away with it).
Driving at a constant speed will not affect your reception - the PLL in the receiver has no problem locking in to a signal that is significantly off the target frequency (much more than the shift you can get from car motion - accounting for all other sources of drift in transmitter and receiver, and for the fact that the modulated signal varies by 256 kHz). 
You might wonder: if you attached your antenna to a wheel that is spinning very rapidly, would you be able to hear a low frequency hum as you add a small modulation of the carrier? Your wheel would have to turn at about 3000 rpm (to give a 50 Hz signal); and if the full amplitude if 75 kHz (corresponding to e.g. 80 dB on your speakers) and your threshold of hearing is 40 dB (really it is lower, but with a spinning wheel nearby you'd be hard pressed...) you can hear an amplitude that is 100x smaller - which would require a peak shift of 750 Hz and thus a velocity of $3\cdot 10^6 \mathrm{m/s}$. At 50 Hz (300 rad/sec to keep round numbers) you would need a wheel that is 10 km in radius.
So you see - there is really is nothing to worry about. Your radio can cope with your driving habits.
A: It does! However it doesn't change the frequency enough to matter.
An FM transmission is not a precise frequency. Instead it spans a range of about 100 or 200 kHz depending on which country you are in. So your FM radio actually accepts a range of frequencies either side of the central frequency.
Let's suppose you're travelling at the maximum speed permitted in the UK, which is 70 mph or just over 30 m/s. This will Doppler shift the frequency of the FM station by a factor of about 1.0000001. In the UK the FM frequency is around 100 MHz, so the shift in frequency is about 10 Hz. This is only 0.01 % of the range of frequencies the transmission uses, so the frequency shift does not affect reception.
To seriously affect reception you'd need to be travelling at around 100 000 miles per hour.
For completeness I should probably add that modern radios auto tune, and would automatically compensate for a change of frequency due to the Doppler shift.
