The stickies cover it in detail, and you should read all the metallurgical stickies, but here is a super simplified description of HT.
The structure (atomic arrangement) of the steel you are working on determines how hard and tough it will be.
When shaping the blade the steel is annealed, and is soft enough to file and shape. All work on shaping is done before HT.
When you heat the steel above 1350F, it becomes AUSTENITE.
When you cool the steel one of two things will happen. 1) If cooled fast enough the austenite will convert to MARTENSITE, which is the hard structure knives are made from. 2) If cooled too slow, the austenite will convert into soft PEARLITE. This won't hold an edge.
After the hardening, the martensite will be very brittle, and the blade will need to be TEMPERED, which makes it tougher and brings the hardness down to a usable point.
In a full quench, the entire blade is heated about 100 degrees above 1350F ( where it becomes non-magnetic), and held there long enough for the austenite to become ready for the quench. 1450F is the normal temperature for hardening most simple carbon steels. The entire blade is plunged, usually tip first, into a tank of quenchant to cool it down fast enough to become martensite. The quenchant needs to be fast enough for the steel type used, but not too fast ,or the blade may crack. For some carbon steels, Canola oil will work. Many new makers use it on their first knives. There are commercial quenchants made for quenching knives, and these are the best choice when you are ready for that step. Some steel types, like 1084, 1095, W1, and W2 need a fast quenchant.
In a partial quench, sometimes called an edge quench, only part of the blade is submerged in the oil. This makes the submerged area convert into martensite, and the exposed area convert into pearlite. There needs to be enough oil to pull the heat away fast enough so the still hot exposed area does not make the edge too soft. A shallow pan of oil won't work well. A long tank, like a turkey roaster, with a piece of screen set on risers so the blade will only go in about 1/2" is what most folks who do this use. Trying to do it by eye will get poor results. After about 10-20 seconds, the whole blade is placed under the oil to suck the rest of the heat from the hot spine.
Some makers use a big torch ( welding torch with a rosebud tip) to heat only the edge and when it is a bit above non-magnetic, do the edge quench.
Some makers like an edge quench and some don't. It is a matter of personal preference mainly, but there is a metallurgical reason for doing a full quench. Martensite is stronger than pearlite. If you want a softer spine and a harder edge, do a full quench and then after the tempering, place the edge of the blade in a shallow pan of water ( a cookie sheet works fine for this) ,and use a torch to heat the spine to near 600F. It will turn blue at that temperature. The edge under the water will stay hard. This will soften the spine a lot more than the edge, and is popular for rugged use camp knives.
Another reason that a full quench can be better is that you have control over the blade temperature at the time of the quench. Heating the blade with a torch to heat only the edge is guess work about how hot it really is ,and if the edge is evenly heated. Heating the entire blade is more accurate and easier to regulate. Also, some steels need to sit at full temperature for up to 10 minutes before the quench ( this allows the alloy ingredients to re-dissolve) and that is nearly impossible to do with a torch.
