Ceramics are brittle materials, rocky I guess you could say. They all share the properties of being relatively strong, hard, brittle (no permanent plastic deformation, unlike metals which are malleable and ductile), and most distinctively, are often melted or sintered.
Making such a definition really isn't possible, since engineered ceramics and pottery cover such a dramatic range; pottery is mostly sintered silica-based oxide materials, while engineered ceramics include non-oxide sintered products such as silicon nitride, single crystal items like cubic boron nitride and diamond, as well as cermets (ceramic-metal composites) like "carbide" cutting tools (which are usually fine tungsten or tantalum carbide powder bonded with cobalt or other metals).
They all basically have the same thing in common (except the grown crystals, of course): they are sintered from a fine power. Well what is sintering?
Say for example my crappy drawing on the left is some material. It is composed of fine grains (usually on the order of microns in diameter), either loosely piled, pressed or bonded together. It could be silicon nitride having been compressed in a steel die, aluminum oxide with an organic binder (which burns out), or clay which has the nice property of bonding itself with the help of water. The particles are basically piled together, having little or no "green" strength. The center picture is a cross-section after sintering: the particles are tighter, the interfaces are wider - bonds have formed - and porosity is much lower. But why?
I like to think of it this way: you've always been told heat is motion, right? Atoms vibrating and such? Well, give them enough heat and they really start to move. They don't move much, but when you have a very fine powder, they move enough that they start to move around into adjecent particles. This is called diffusion. This is accompanied by a reduction in center-to-center distance between the particles, which causes shrinkage. Uh, I'm not really sure why this is; I don't think diffusion alone accounts for it. Or perhaps it does, causing the atoms to draw together into a tighter structure, basically a really slow surface tension. At any rate, the pores close up and it becomes dense, bonds form, thicken and for very fine powder, pressed while being heated near the melting point, becomes almost fully dense - without having been melted.
There is also another method, known as liquid phase sintering. You may already be familiar with this - say moisture gets into your salt and it turns to a uselessly solid cake. Well, salt is soluble in water, so that when the water gets in, it dissolves some salt and becomes a saturated solution. On removing the water, the salt gets left behind. If the salt is suspended in this water (as a slurry for visualization purposes), the particles are seperated by the water. As the water level decreases, there is less to seperate the particles and it becomes more dense. Additionally, the solution becomes supersaturated, releasing salt, depositing it on nearby crystals, causing them to grow. When the water finally leaves, the particles end up bonded to nearby particles, and since this happens throughout the material, you have an annoying plug of salt! The water can be removed a few ways, evaporation is one. It can also be adsorbed into surrounding materials, or even the salt itself. This is common in deliquiscent materials like anhydrous calcium sulfate (hence plaster of paris!). Or, in the case of a melt solution, a eutectic mixture forms between the grains, first partially melting them, then depositing material on the grains, then on cooling it solidifies in place, increasing bond strength and reducing porosity. Since oxide crystals form very slowly (quartz is an extremely complex structure!), this eutectic is often a supercooled liquid, otherwise known as glass, in pottery. You can imagine liquid-phase sintered pottery as the far right diagram above, hard particles (which are either insoluble, or were not melted during the firing) suspended in a glassy matrix.
Pottery, in the purest sense, is composed of clay. I just gave a big dissertation on sintering, so you are expecting them to be sintered. Fine particles sinter well, so let's start there: a good bentonite is made of extremely fine particles (colloidial in fact), so sinters very well on its own, unfortunately it also shrinks very well on its own, both due to drying (the small particles disperse evenly in the water) and shrinkage on firing. In practical terms, it tends to be impure with fluxes (a byproduct of its colloidial nature, making it hard for nature to seperate the insoluble clay suspension and soluble fluxes), so matures at too low a temperature (although is isn't much of a concern when heat is costly). As a result, most pottery is done with a kaolin or ball clay instead (with bentonite reserved for improving green strength or plasticity, both due to its fine grain size).
Kaolin is a primary clay, meaning the deposit was formed in situ by weathering of feldspar type minerals (granite, etc.). It also tends to be coarse (as clays go), so has relatively low shrinkage but also doesn't mold well (low plasticity). Ball clay is a secondary clay, meaning erosion worked away at one of those kaolin deposits, washed it downstream, then the stream widened, depositing it. The coarser particles fall out upstream, leaving a finer clay. Ball clays, by definition, are usually pretty pure, but not as much as kaolin. General practice is that both are acceptable, with ball clay usually added to improve plasticity; plain ball clay is very sticky stuff, while plain kaolin is reasonable for molding.
That covers particle size, but obviously, that's not everything. Let's look at two things next, composition and melting. I don't know what you have, but I know I get KT#1-4 ball clay locally for a fin per 50 pound bag. The composition can be found in DigialFire's database (along with a short blurb and some other typical data) and is listed as:
The definitions can also be found at DF, matter of fact a great wealth of information is available but I'll summarize what we need to know here:
The fluxes will combine with varying amounts of the other materials, for argument's sake let's say the 2% flux content forms a glass with 10 times its weight (not unreasonable - soda-lime glass is about 20% fluxes); 20% of the body melts. This will melt around cone 10 (2100-2300°F depending on speed), leaving the 70% refractory content (minus 10% LOI, mind you) as a framework surrounded by sticky glass. On cooling, the particles have diffused together a little, and more importantly, they are bonded with this glass. Especially since ball clay is fine, it ends up very strong. I once fired a half inch thick slab of raw clay, intending to grind it down into grog - fat chance! I dropped that thing on the concrete floor and it didn't break. Eventually I heated it orange hot then dropped it in a bucket of water, microfracturing it.
Another understanding of sintering can be had with a phase diagram. Starting from the left at 100% SiO2, we first have quartz / cristobalite / tridymite phase (whichever it feels like crystallizing as, or more likely given the "fast" firing rates, a glass). The blue line is the liquidus point, above which all is liquid and below which, at least one phase is solid. (A mixture of liquid and solid is a "mushy state".) The left area is SiO2 + L, center is mullite + L and on the far right, Al2O3 + L. The areas below these are fully solid, and are labeled. On the left, silica; middle, silica + mullite (a mixture of the two), near right is mullite itself (3 Al2O3 + 2 SiO2) and further right is mullite + alumina. The horizontal lines seperating the partially liquid and fully solid zones are solidus lines. Where the liquidus and solidus lines meet is called a eutectic, the lowest melting point composition of any particular combination.
My clay fits in around 32% SiO2 (referring to the ratio of alumina and silica alone), which you can see starts melting at 1600°C (2900°F) and is fully molten around 1800°C (3270°F). This is about right considering KT#1-4 is listed as PCE 32, meaning if you use a sample as a pyrometric cone, it will fall over at cone 32 (about 3120°F). The theoretical temperature is about cone 36, the difference being due to the flux content. Still not bad, eh?
So what happens during sintering? As can be seen by the melting point, the flux content isn't all that significant, and that's deeply into the white hot realm. At lower temperatures, almost no melting will take place (especially below 2900°F, where the SiO2/Al2O3 eutectic melts), so we're stuck with diffusion. It turns out this works just fine, as I noted above with my grog slab, and industry proves daily with diffusion-bonded materials.
Because strength at low temperatures and appearance are the only concerns, relatively low firing (compared to a good cone 20 needed to liquid phase sinter KT#1-4) is often used. There are some limitations with various materials that require high firing for certain characteristics, but for the most part, clay blends are vitrified by cone 6 or 10. How do they do this? Extra fluxes. By definition, a flux lowers the melting point - the original flux, fluorite, was named for the latin "to flow". Fluorite (calcium fluoride CaF2) isn't actually used much in pottery (more often in metallurgy, which makes it odd that I don't have any), but its oxide equivalent is. Calcium oxide is a pretty strong flux, active above yellow heat. Just fold in some powdered limestone or dolomite and your clay will vitrify much sooner. A little soda or potash (often sourced from something insoluble like feldspar, which is a good source of calcium in certain varieties too) helps spread out the flux content, something that isn't really necessary, but works better for some reason. I guess. Feldspar is preferred because it is naturally "mixed" so has a low melting point, whereas mechanically blended lime has to be dissolved by heat action before it'll form much of a solution. Anyway, now that you are melting things, a glass phase is forming and the word is "vitrified". At room temperature, this glass is hard and strong, but kind of useless for me since I need strength at high temperature. And so ends this broken, rambling page. For today.