If an antenna must be $\frac{1}{4}$ of the wavelength, how can car antennas be so small? If the transmission antenna has to be $\frac{1}{4}$ of the wavelength, how can the car antennas' size  be much less than that and properly receive the radio signal? 
 A: With a resonant antenna, the reactance (capacitive and inductive) should be zero.  Short antennas are usually capacitive so that capacitive reactance is offset using an inductor.  Often for an AM radio a loop inductance is included.  Also, some antennas are longer than they appear because the conductor is wrapped around the core of the antenna (sometimes you can see this).
And, note that a short antenna is an inefficient antenna but often AM signals are so strong that the poor efficiency is not an issue.  The antenna length is much closer to the 1/4 wavelength needs of the FM frequencies.
To get the full 411 on antenna design and matching issues, consider taking a look at the most recent ARRL Antenna Book.  I think the most recent is the 22nd edition.  You might find it in your local library or you can buy a copy from the ARRL store at http://www.arrl.org.
A: It doesn't really matter if you're talking about transmission or reception; the answer is basically the same. A quarter-wave antenna for the commercial AM band would be about 80 meters long, which would clearly be impractical for a receiving antenna. The gain is lower because it's not resonant, but that's OK, you just amplify. You can amplify a lot because although noise is also picked up, the antenna is equally insensitive to noise and signal, so the signal-to-noise ratio is still OK after amplification.
A: The idea behind the quarter wavelength antenna is that it is self-resonant: it is "tuned". You can however use an antenna of any size to pick off some electromagnetic energy - and you can tune the antenna by adding some inductance in series (or inductance and capacitance). The reason that you tune an antenna is simply this: you want it to have real impedance, which happens exactly at resonance. When this happens, then all the power that is incident on the antenna ends up going into the electronics; when the impedance is complex, the current either lags or leads the voltage, and this results in reflection of the power (and less power going into the amplifier). 
You can't get around the fact that you are "capturing" energy from less area in space, so it will be less efficient. On the other hand having an antenna that is too long doesn't buy you anything as the signal from one part of the antenna would cancel the signal from another.
This is one of the beautiful things about the Yagi antenna: the multiple elements are spaced in such a way that they cause constructive interference along a particular axis, giving you both gain and directionality - in essence you are capturing energy from a larger volume of space.
Little anecdote: from wikipedia article on Yagi-Uda antenna

The Yagi was first widely used during World War II for airborne radar sets, because of its simplicity and directionality. Despite its being invented in Japan, many Japanese radar engineers were unaware of the design until very late in the war, partly due to rivalry between the Army and Navy. The Japanese military authorities first became aware of this technology after the Battle of Singapore when they captured the notes of a British radar technician that mentioned "yagi antenna". Japanese intelligence officers did not even recognise that Yagi was a Japanese name in this context. When questioned, the technician said it was an antenna named after a Japanese professor.

I have heard it said that the inventor was executed during WW-II by the Japanese because his invention "helped the enemy". I can't find a reference to support that. Apparently he wasn't even the inventor - it was Uda. But Yagi published it, patented it, and even sold the rights to Marconi (before the war). It would be karma if he ended up paying the ultimate price for stiffing the real inventor...  "No, it wasn't me! But Professor, your name is on the patent... off with his head." In fact it is apocryphal - a bit more searching reveals that he died in 1976.
A: Antennas don't have to be sized to the wavelength of the signal. It just happens, that if they are significantly smaller than a quarter wavelength, the sensitivity drops very quickly with antenna size. That, by the way, is also an enormous advantage for electronics design of switched mode power supplies. If the transmission efficiency wouldn't drop, our switched mode power supplies would need very expensive shielding to prevent them from being efficient transmitters. As it is, we can get away with very little, very cheap shielding because the circuit sizes are small fractions of the antenna size required at those frequencies. 
A: Antennas don't have to be sized to the wavelength of the signal. It just happens, that if they are significantly smaller than a quarter wavelength, the sensitivity drops very quickly with antenna size. That, by the way, is also an enormous advantage for electronics design of switched mode power supplies. If the transmission efficiency wouldn't drop, our switched mode power supplies would need very expensive shielding to prevent them from being efficient transmitters. As it is, we can get away with very little, very cheap shielding because the circuit sizes are small fractions of the size that would be required of an efficient antenna at those frequencies. 
There is, by the way, absolutely no difference between receiving and transmitting antennas. The efficiency hit is the same for both types. 
