Both ceramic resonators and quartz crystals work on the same principle: the vibrate mechanically when an AC signal is applied to them. Quartz crystals are more accurate and temperature stable than ceramic resonators. The resonator or crystal itself has two connections. On the left the crystal, right the ceramic resonator.
Like you say the oscillator needs extra components, the two capacitors. The active part which makes the oscillator work is an amplifier which supplies the energy to keep the oscillation going.
Some microcontrollers have a low-frequency oscillator for a 32.768 kHz crystal, which often has the capacitors built-in, so that you only need two connections for the crystal (left). Most oscillators, however, need the capacitors externally, and then you have thee connections: input from the amplifier, output to the amplifier, and ground for the capacitors. A resonator with three pins has the capacitors integrated.
The function of the capacitors: in order to oscillate the closed loop amplifier-crystal must have a total phase shift of 360°. The amplifier is inverting, so that's 180°. Together with the capacitors the crystal takes care of the other 180°.
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When you switch a crystal oscillator on it's just an amplifier, you don't get the desired frequency yet. The only thing that's there is a low-level noise over a wide bandwidth. The oscillator will amplify that noise and pass it through the crystal, upon which it enters the oscillator again which amplifies it again and so on. Shouldn't that get you just very much noise? No, the crystal's properties are such that it will pass only a very small amount of the noise, around its resonance frequency. All the rest will be attenuated. So in the end it's only that resonance frequency which is left, and then we're oscillating.
You can compare it with a trampoline. Imagine a bunch of kids jumping on it randomly. The trampoline doesn't move much and the kids have to make a lot of effort to jump just 20cm up. But after some time they will start to synchronize and the trampoline will follow the jumping. The kids will jump higher and higher with less effort. The trampoline will oscillate at its resonance frequency (about 1Hz) and it will be hard to jump faster or slower. That's the frequencies that will be filtered out.
The kid jumping on the trampoline is the amplifier, she supplies the energy to keep the oscillation going.
Further reading
MSP430 32 kHz crystal oscillators
Usually, yes - assuming you mean 32.768kHz when you say 32kHz, or the watch will run at the wrong rate.
First you need to know which IC pin is the input and which is the output, as Ignacio commented. A quick measurement with your oscilloscope will show a strong 32.768kHz signal at one pin on the crystal, and a weak signal at the other. The weaker signal is the input.
Now you need to connect the watch ground to the DS3231's ground, and capacitively couple the DS3231's 32.768kHz output into that input, removing the crystal of course. The correct value of capacitance is a guess, but 22-100pF is probably the range, and I would guess 33pf as a starting point. After the capacitance, you should see a signal approximately the same as the original. If it's much larger, reduce the capacitance and vice-versa. (You may have to remove the trimmer capacitance, or turn it to its minimum setting, i.e. maximise the input signal)
Oh and you still need the correct supply voltage of course - e.g. the original battery.
There should be a good clean copy of your 32.678kHz signal on the original (now unused) output pin.
Best Answer
Quartz crystal is not the same thing as crystal oscillator.
Quartz crystal consists of real physical crystal that has precise dimensions. The higher the desired frequency, the smaller the crystal must be. So it is practical to make them only in certain range (common values are 4 MHz - 40 MHz, and then low-frequency 32 kHz for RTCs).
It works by piezoelectric principle. Applied voltage causes it to contract and vice versa. At certain frequency, it gets into mechanical resonance which is projected into electrical resonance.
Quartz crystal oscillator is circuit that contains a quartz crystal and produces periodic signal. It can be integrated or constructed from individual components. One example is single-inverter oscillator.
The part numbers you provided are almost certainly form an 4MHz crystal.