Once again, if you want faultless, extra-safe directions, do not listen to me. Go listen to the nice folks at NCHFP. Thank you, your public service announcement is now over.
Okay. You've made your applesauce or jam or whatever. (Note from sad experience: If you're canning fruit, simmer it briefly in its syrup or juice, strain it out with a slotted spoon and pack tightly into jars, then cover with syrup. You'll be much happier, trust me.) It's still VERY HOT. (If it is not very hot, bring it back to a boil.)
Ideally, do this before you boil your stuff-to-be-packed:
Take your giant pot and fill it about 1/3 full of hot water. Set it on high while you do all the rest of this stuff. If you're sterilizing jars, take a smaller pot (big enough to dip your glass jars in), bring water to a rolling boil, and dip all your jars very carefully so that the insides get coated in boiling water and you do not get burned. Set them upside down on a clean dishtowel on top of something heatproof (those tiles are useful here).
For jam, you want to actually sterilize your jars, because boiling pectin too long can make it hydrolyze. If I'm canning anything with a 15 minute+ processing time, including applesauce and tomato sauce, I don't bother. Your jars need to be clean.
It's a nice idea to put all your jar lids and rings in a strainer and dip them in boiling water for a minute or five. If you're doing jam and have bothered to sterilize your jars, then yes, boil your jar lids and rings. Otherwise, whatever.
Hot pack your stuff into jars.* That is: Pour your really hot whatever into your nice clean jars, using your handy canning funnel. Leave AT LEAST A HALF INCH AT THE TOP- maybe more- not above the jar's rim in any case. Otherwise the contents will bubble up and get everything sticky.
Wipe the rims with a clean, damp cloth. Put on a clean, new lid and a ring. Screw it on as tightly as possible.
By now your water in your giant pot is hopefully heading for boiling. For jam, wait until it is at a rolling boil. For everything else, go ahead and put it in. The water needs to be at least a half-inch over the tops of the jars. If it's not, add more boiling water (a teakettle is helpful here).
When your water is at a rolling boil again - it usually takes a few minutes- start timing it. Look up your processing time at the NCHFP. Jams in sterilized jars need five minutes processing; in non-sterilized, ten. Applesauce is generally 20 minutes for quarts (15 for pints). Add a minute per 1000 feet above sea level. (You can turn the stove down a few notches as long as it keeps boiling.)
Take your jars out one at a time, with your jar tongs, and put them on your dishtowel on a heatproof surface, lid side up (never turn upside down). As they cool, the lids should pop inwards. Check them all 3 hours later (or in the morning) and, if it's still hot, put unsealed jars in the refrigerator. If it's the next morning, throw it away. (I almost never have jars that don't seal).
Store canned goods in a dark place; they really will photo-oxidize otherwise. This isn't dangerous, but it affects color and flavor. Refrigerate after opening. Most things keep at least two years (Official Sources usually say one year, but I eat them for 2 or 3), or longer, but the flavor usually goes downhill after two years.
*Sole exception: fermented pickles. If you're canning fermented pickles, you probably do not need to be reading my blog post.