Canning: A Relatively Brief And Somewhat Vague Tutorial, By Request (Part 1)

If you want a really good, detailed, relatively brief tutorial, see here (pdf). The National Center for Home Food Preservation is where I always look up processing times (how long am I supposed to boil this, again?)

But, by request, here's my summary of How I Can Stuff, By J.F., Age 32.

(If you're wondering how I make applesauce, I bake and/or boil-on-stovetop whole apples with about a quarter-inch of water, until they explode, and then run them through a vertical food mill (this is the one I use; it's perfectly adequate and also inexpensive), season as desired, and process. PRO TIP!  Run cooked apples through a food mill while steaming hot - you'll get much better yields, and won't ever have to run it through twice.

Things you really truly need to do canning:*

Heatproof glass jars, sold as Mason/ Bell jars in the US. (These can often be found at Goodwill-type stores and garage sales.  Check the rim for chips.  They're fine to re-use as long as the rim is intact.) Various Official Sources say to never re-use commercial mayo/whatever jars, even if your Ball lids and rings fit, but I've not yet had one explode.
As many rings as you have jars.
As many CLEAN AND UNUSED lids as you have jars.  Never, never re-use lids, please!  They're 20 cents each!  Trust me, getting botulism is not worth even many, many packets of new lids.**
A jar tongs. No, really, you need this.  Trust me.  Regular tongs will leave you with scald marks, and also broken glass with jam in all over your kitchen floor.
A really big pot.  The rack is optional, but nice, to prevent stuff from banging around.  If you don't have one, put a ratty kitchen towel in the bottom.  You need at LEAST two inches in your pot above the top of your tallest jar.  (That is, the pot should be tall enough that jar + 1/2 inch to 1 inch water on top does not lead to hot water boiling over everywhere.) Seriously, put the jar in to check, or you'll regret it later.  Please don't pay more than $20 for this item, because both Amazon and Walmart have them for $20.
Some clean dish towels.
A heatproof surface to put all your really hot canned stuff on (the cheapest of 18 x 18 tiles from your local hardware store work great).
A time-keeping device of any kind.
(If you are in the mountains, a knowledge of your altitude.  Add one minute processing time per 1000 feet above sea level.)
If you're doing more than a few jars, a canning funnel of some kind.  They're like $1.50 and will save you an hour per batch in wiping sugary goo off every surface in your kitchen.  Really.

Things that are nice:
A strainer (the mesh kind with a wooden handle)
A regular tongs of some kind
A fan to blow on you while you're in a hot steamy kitchen
A butterknife, if you have the patience to de-air-bubble your canning stuff.

Things that are completely unnecessary:
A lid lifter.
A bubble freer.

Next: How To Actually Can Stuff

*These are not affilitate links.  Amazon is merely convenient. Walmart also stocks all these things (at least around here).  
** Our last ER trip (not for botulism, but still) cost $289.50. This is 115 packets of jar lids.