What makes quartz oscillators vibrate? I understand that the principle behind it is piezoelectricity and electrostriction (inverse piezoelectricity), but how does one make the crystal vibrate?  
The only thing I can think of is using an alternative tension source, but wouldn't that make the crystal vibrate at the frequency of the source?
Also, for example quartz wristwatches use a battery as a power source, which are DC. 
From what I understand, it seems like it is sort of cyclical system where the where an initial tension create a strain via electrostriction, and that strain causes a piezoelectric effect which creates an electric field, which creates electrostriction... In that case, how would the battery help the system?
 A: Electrostriction is the property of insulators that makes them change their shape when an electric field is applied. Piezo-electricity is the reverse: it makes the material generate an electric charge when force is applied.
Suppose we apply a varying electrostatic field to the crystal. As the voltage goes up and down, the crystal will compress and expand. This change in shape in its turn generates an electric field, which interferes with the applied field. When the two are in sync the output from the crystal will be maximum, and the crystal will be vibrating with its maximum amplitude. 
The electronic oscillator circuit that generates the varying field is designed to use feedback to adjust itself to the frequency that generates the maximum output, and as a result its frequency is locked to that of the crystal. For this to work, the starting frequency needs to be reasonably close to the resonant frequency of the crystal, but that does not present any real problem.
A: I answer you without knowing the details of quartz oscillators for clocks, but on general principles and having seen some quartz oscillators.
You indeed use an oscillating tension source, and then you measure the oscillations that you induce (I think this is made by measuring the current). The oscillator is strongly resonant at its natural frequency, so you can look for the frequency at which you have the strongest response (that is, the highest value of the current). In this way you determine the frequency.
This is also a general way to use oscillators to mark time: excite them with an oscillating input, find the frequency at which their oscillations are largest. Take a look at this Wikipedia article for a general idea.
I do not have in mind the details on how energy is dissipated in the cyclical system in which the tension is transformed into a mechanical oscillation and that back into tension, but there must be some loss somewhere, and the battery will fill in the dissipated energy.
A: There are some confusions: Electrostriction is not the inverse or converse effect of piezoelectricity. Piezoelectricity appears as direct and converse linear effect. Moreoveor: "Electrostriction applies to all crystal symmetries, while the piezoelectric effect only applies to the 20 piezoelectric point groups. Electrostriction is a quadratic effect, unlike piezoelectricity, which is a linear effect" (Wikipedia).
A: I thnk, this question woul better fit into electronics.
But you're absolutely right. If whe apply a DC Voltage to the Quartz ist will just deform. But while deforming, it changes it's electrical properties.
From an electrical standpoint, a Quartz has a capacity, an inductance and a resistance.
If we apply a positive edge at one end (let's call it "input"), the level at the other end will rise end (let's call it "output") will rise as well, but with a certain delay. Now we just feed that output signal into an inverter, which will "flip" it's output to low once a certain treshold at the input is reached. Now feed the output of the inverter to the input of the quartz and you have an oscillator.
So yes. A quartz doesn't oscillate "just so". It's basically just an element, to keep the freqency of an oscillator very stable (and it's astonishing to what precission they are stable considering the price).
