Metals deform plastically by means of shear between atomic layers. However, shearing an entire plane of atoms simultaneously takes proprotionally enormous stress. Instead, a half-plane of atoms perpendicular to both of the shear direction and shear plane gets pushed about one atom radius away from the shear plane, forming a dislocation at the shear plane. The dislocation is easily moved by application of shear stress in a manner likened to that of an inch-worm. As dislocations move, they get caught on other defects (especially other dislocations, grain boundaries, and hard inclusions) and require greater stress to keep them moving. While the full atomic level process that causes work hardening is not fully understood (though there are theories such as dislocation pinning and Frank-Read sources), dislocations increase in density with increasing plastic deformation. As they increase in density, they tend to run into each other more, requiring ever greater stress to continue deformation, resulting in strain hardening. Consequently, with increasing plastic deformation, or work, ductility decreases and yield strength increases.
Dislocations deform crystals locally producing strain fields, which store energy. As a result, dislocations increase the free energy available in a crystal. Free energy drives spontaneous processes, which in this case is the reduction of strain energy by removal of dislocations. Metal crystals also have vacancies, or missing atoms, which can interact with dislocations to move them to the surface of the crystal where they are readily annihilated. Dislocations may also annihilate each other if two opposing (up vs. down) half planes meet. Increasing temperatures causes dislocation removal processes to occur exponentially more rapidly, eventually occuring within minutes or even seconds depending on the metal and the section thickness. As a result of dislocation density reduction, the properties of the metal change opposite that of work hardening: yield strength decreases while ductility increases.
Tempering is a mostly unique process in that it is almost exclusively used with respect to martensitic steel. The term is also used with other materials such as glass and some high-performance polymers. If a hardenable steel is quenched rapidly enough the microstructure that forms contains martensite. Martensite forms as a result of carbon solubility in the austenite phase of steel at elevated temperatures. Ferrite has very low carbon solubility, so as temperature decreases carbon proceeds to diffuse out of the iron-rich phase. If the cooling rate is slow, the carbon is able to diffuse out and form cementite. If the cooling rate is sufficiently rapid, the diffusion process is too slow and the austenite transforms to martensite instead of ferrite, trapping carbon in the martensite crystals. Ferrite has a BCC lattice, while martensite has a strained BCT lattice due to the trapped carbon atoms. The sudden volume change of the transformation, creates internal stresses in the grains causing rapid, intra-grain work hardening. As a result, the strength of the martensitic steel is high and the ductility low, making it brittle. Tempering is effectively an annealing process, reducing the effects of the transformation-induced work hardening, and restoring some ductility. Interestingly, a carefully crafted tempering schedule can restore significant ductility without significantly reducing strength. The restored ductility is due to the reduction in dislocation density, while the maintained strength is due to the carbon-strained BCT lattice. It takes more stress to move a dislocation through a strained lattice. If the tempering goes on too long or at a too-high temperature, the carbon will diffuse out of the martensite, forming ferrite and pearlite, and return strength and ductility to that of a non-martensitic steel.
Quenching is useful for producing martensite, as noted, but may also be used to produce other microstructural components including bainite, pearlite, and combinations of all of the above, according to TTT diagrams, also known as isothermal transformation diagrams. It is possible to schedule heat treatments to take advantage of quenching by rapidly reducing the temperature of a steel component to a specific temperature and holding it there. Such a quench and hold treatment allows one particular microstructure, as noted on the TTT, to form instead of the others. Naturally if such a temperature is higher than the boiling point of water, another medium such as heated oil must be used. By quenching, atomic mobility is decreased exponentially, preventing or slowing diffusion-based processes. The formation of pearlite and bainite are both diffusion based, while martensite is diffusionless. Specifically, martensite requires there to be minimal carbon diffusion.
Carburizing is a diffusion-based process by which carbon is implanted in a metal at elevated temperatures. By now you can probably guess that elevated temperatures make the process go exponentially faster. The process can also be controlled based on the external carbon source concentration, and the amount of time spent processing. A carbon gradient forms in the metal, with the highest concentrations at the surface and decreasing inward. As noted in the tempering section with martensite, higher carbon content distorts the iron lattice, increasing internal stress, hindering dislocation mobility and increasing strength. Higher strengths are also associated with higher hardness and ultimately higher wear resistance. Note however that martensite does not form as a result of carburization, instead the existing BCC lattice becomes highly strained. Parts such as gears are often formed from a relatively low carbon steel and then carburized. This gives a shell of hard, wear resistant steel surrounding a ductile, damage-tolerant core which limits through-cracks. That way the surface may wear and chip, but the entire gear won't crack. A related process is nitriding, using nitrogen instead of carbon. Some metals exhibit extreme brittleness with low concentrations of one or both materials, so not all metals may be strengthened this way.
While most of the above has been applied to steel, these concepts may typically be broadly applied. Work hardening and annealing apply to all sufficiently impure metals at and above room temperature, with a few exceptions (gallium and mercury spring to mind). "Pure" metals also exhibit unusually high ductility compared with metals containing even slight (>0.1%) impurities, as do single crystals. Six-nines fine gold can be worked endlessly at room temperature as the dislocations are annihilated as fast as you can create them. Pure iron anneals itself over the course of a few minutes at room temperature. However even A 0.01% carbon impurity is enough to raise the one-hour-per-inch annealing temperature to hundreds of Celsius.
Tempering generally only applies to martensitic steels in the realm of metals, as far as I am aware. Some nickel alloys are capable of forming a diffusionless martensitic phase, and if they were widely used commercially I imagine tempering would be applicable to them as well.
Quenching is applicable to any metal that requires rapid cooling as part of a heat-treatment schedule.
Carburizing applies to any metal for which it may be desirable to implant carbon atoms in the crystal lattice. Aluminum and gold are not good candidates for carburizing as they have virtually no solubility. Nitriding is the same process as carburizing but with nitrogen. Other non-metal elemental gases are generally deleterious for all properties, especially oxygen and hydrogen. Hydrogen makes virtually every metal extremely brittle, and oxygen oxidizes metal.