I’m having trouble grasping the true nature of temperature. Can someone please explain how it emerges fundamentally? Wikipedia unfortunately was a bit of an overload on the subject. Is there a simple analogy to this phenomenon?
Temperature measures the average amount of kinetic energy in a substance. As I am sure you know, everything that exists is made of molecules that are bonded together. What you may not know is that those molecules continue to move and vibrate. The more kinetic energy, the more vibration and movement, the higher the temperature. If a solid starts to vibrate at a high enough frequency, some of its bonds break and it becomes a liquid. The liquid can continue vibrating at higher and higher frequencies until more bonds are broken and it becomes a gas. Anyway the amount of vibration and movement is directly correlated to how much kinetic energy is available. I hope that makes sense.
An important distinction: Temperature is a measure of the amount of kinetic energy per particle. If I increase the number of particles (typically molecules), I’ll increase the total amount of kinetic energy, but I won’t change the temperature.
Note also that this definition of temperature is a grave oversimplification. For starters, the proportionality constant between temperature and energy-per-particle can be different for different sorts of materials, and some things can have temperature without even being composed of particles at all (like black holes). But you probably don’t want to hear the full, technically-correct definition.
Is there something about these bonds that determines the phase? Say you had just 2 water molecules. Can we examine those in some way to determine “these are solid” or “these are now a liquid?”
Roughly speaking:
solid - particles are vibrating back and forth in a sort of lattice or other regular arrangement, none of them changing place, just bouncing off each other and humming in place
liquid - particles are each always sticking tightly to several others, but moving in odd currents, each in its own direction, so one particles neighbors are constantly shifting. Like a crowd of people mingling at a cocktail party, in a room that’s really too small for all of them to have personal space. 
gas - particles are flying through space and only occasionally bouncing off each others or meeting other particles. (Roughly speaking - in a fairly dense gas that ‘occasionally’ might happen several times a second at least for a particular particle, but it’s not CONSTANTLY in contact with others like in a liquid.)
That’s the way I used to remember it, and I believe that it’s fundamentally close to correct, quantum uncertainties aside. 
Note that these are usually behaviours that would be much easier to observe with many particles than just two water molecules, especially the distinction between solid and liquid.
You might. If the added particles have a lower average than the ones being added to, then you’ll lower the temperature. This is exactly what happens when you pour a cup of ice water into a pot of boiling water.
It should also be noted that temperature is a perceptual quality, a human concept. All techniques used to measure temperature do so indirectly, by measuring something like the length of a column of mercury or expansion of a metal coil that is reacting to the kinetic energy in the air or other thing that it touches. We just calibrate it to be consistent with our concept of temperature.
It’s similar to color. Light is a spectrum of EM frequencies; the color “red” is a subjective perception of a certain band of those frequencies. Sort of like that Zen koan about a tree falling in the forest.
Temperature is sometimes described as the intensity of heat. This is not the same as the amount of heat. For example, the flame at the tip of an acetylene torch may have a much higher temperature than the flame on a stove burner, but the stove puts out more total heat and will boil a pot of water faster.
I’m sorry, but this make no sense at all. Temperature is a repeatably verifiable physical property of a material. Perception has nothing to do with it.
Similarly, a bucket of water at 0 C contains a lot more heat than a cup of water at 100 C.
So it sounds like temperature is the average of the amount of kinetic energy per particle in a whole bunch of particles. Can one particle have a temperature? If not, then how many particles are required to produce a temperature?
With some things, it makes sense to talk about them having two different temperatures at the same time: Plasma
I wouldn’t say that you have a single fluid with two different temperatures, there: I would say that you have two different fluids, each with its own temperature, which just happen to be occupying the same space.
Well, if you were to talk about the temperature of a single particle, then most folks would probably get roughly what you meant, but properly speaking, it’s a concept that only applies to systems of many particles. What exactly constitutes “many” is a very fuzzy question, though.
If that’s true, why does the temperature of a gas increase when it is suddenly compressed (i.e. adiabatic compression)? The number of particles doesn’t change, and little or no energy is added to the gas (the definition of an adiabatic process is that no energy is added to or lost from the system, but this isn’t achievable in the real world).
I would expect temperature to be a measure of kinetic energy per unit volume, not per particle.
Temperature the physical property is also called ‘temperature’ as a perceptual quality. The reflection spectrum of something represented as a function of intensity per wavelength confined to the visible spectrum is also a repeatably verifiable physical property of a material, but in the perception domain it’s called “color”. Makes sense to me.
1.5 times as much, in fact, assuming a 1-gallon bucket and an 8-ounce cup.
I think of temperature as the ability to transfer heat to (or from) other objects. I really don’t think you can define it more narrowly. The sky has a well-defined temperature, even though you can’t measure or define the motion of particles there.
In an adiabatic process, no heat is added to or lost from the system. But heat is not the only possible method of energy flow. When you push the piston down to compress the gas, you’re moving against a force, and therefore doing work on (and therefore adding energy to) the system.
Note that while “heat” is often used to mean “the internal thermal energy of a system”, this common usage is incorrect. Heat is one way in which the energy is transferred. There’s no shorthand word for the internal thermal energy.
OK - I’ve thought it through with my dad’s help (he has a degree in mechanical engineering) and have figured out that you’re right - temperature is indeed energy per particle. We thought about what happens in an elastic collision between particles - if they have the same energy before the collision they’ll have the same energy after the collision. This means that two objects with the same energy per particle won’t exchange energy when the particles collide - in other words, they’re at the same temperature.
This makes me curious about something - since temperature can be expressed so simply in terms of fundamental units (e.g. joules per mole), why do physicists use the Kelvin as a unit of temperature? Would it simplify any formulas to express temperature this way?
What you’re proposing is basically to set the universal gas constant R=1, or (since counting in terms of moles instead of units is just a historical vestige) to set Boltzmann’s constant k=1. This would just replace RT (or kT) with T in formulas.
This is sometimes done. Actually, it’s more common to use the inverse temperature beta=1/(kT), because the thermal energy kT commonly sets a scale by which other system energies are measured, giving lots of expressions of the form E/(kT) which can then be written as beta*E. The disadvantage in doing this is that kT is very small at normal temperatures (about 4E-21 J at room temperature), so these units are unwieldy for measurement and communication. This is the same sort of tradeoff as in setting c=1 in relativity; replacing tau=ct simplifies the formulas, but for sizes commonly dealt with by people it’s more convenient to write, e.g., 1 second than 3E8 meters.
This is the right idea.
As a nitpick, the relevant quantity is actually not “energy per particle” but “energy per degree of freedom” (even this is a slight simplification; quantum mechanics limits what should be counted as a “degree of freedom” for a given system at a given temperature). This is why multiatomic but otherwise nearly-ideal gases have larger heat capacities than monatomic ideal gases: they have extra degrees of freedom associated with rotational modes and, at high temperatures, bond vibrational modes. This more general definition also lets one understand the meaning of “temperature” for other systems, such as the 2.7K night-sky radiative bath mentioned by scr4.