Of course the most important book, ostensibly the classic collection of American cuisine, which I refer to most often just as "The Tome," is The Joy of Cooking by Irma S. Rombauer and Marion Rombauer Becker. This is the 1997 reprinting of the 1975 edition, ISBN 0-452-27915-1, NOT the 1997 revised edition, ISBN 0-684-81870-1 (where Ethan Becker tacks his name on to the byline as well).
The 1997 revised edition is an alright book, but it embraces a lot of processed or packaged food which the previous editions do not. It also puts much more emphasis on adding meats or meat products in to dishes where, honestly, they do not really belong or contribute meaningfully. This does reflect more recent trends in American cooking, to be sure, but it makes the recipes less useful for those who do not wish to consume those ingredients. What the revised edition does contribute is much more information about ingredients overall, which I do occasionally find useful. I will admit to having made a few things out of the revised edition, but by and large, it is not the book I reach for when I want to know how to cook something. The notable exception is its recipe for mushroom ragout, which I pour over a Yorkshire pudding. The pages have gotten wet so many times, the book opens naturally to this page.
I would suggest to anyone who wants to learn how to cook better to read The Joy of Cooking cover to cover, whether relevant or not to one's personal tastes, and then do it again. There is, for instance, a section on skinning squirrels, which I can't imagine many people will end up having to utilize (and which was excised from the 1997 revision), but is nevertheless interesting from a food history standpoint. And who knows, maybe you will be in a plane crash in the mountains and have to eat squirrels?
For Japanese cooking I have two books, both by Shizuo Tsuji (Tsuji Shizuo?). The best is Japanese Cooking, A Simple Art, ISBN 978-4-7700-3049-8. I can't even describe how essential this book has been to understanding the basic techniques of Japanese cooking. It presents, largely, very simple, basic "home cooking" sorts of preparations, and has a lot of instructions on garnishes, sauces, and condiments. It does NOT include, unfortunately, a lot of "junk food" recipes, such as takoyaki, okonomiyaki, or yakisoba.
Complementary to this book I think is the more colorful, approachable book Practical Japanese Cooking, Easy and Elegant, ISBN 0-87011-762-9. When I first read this book nearly 10 years ago, I struggled to see what was practical, easy, or elegant about any of the recipes in this book, but honestly for the most part things are presented in this book a lot more simplified and manageable. What this book does is include some more "modern" influences, working in some foreign styles and tastes in a format which is honestly more representative of modern Japanese cooking overall. Nevertheless, it remains more or less "traditional" in its overall presentation. Its recipe for kitsune-udon is the one I still make, every time I make it.
For Southern/French cuisine, a relative newcomer in my collection is Justine's--Memories and Recipes, ISBN 0-9667324-0-5. Some of the best restaurants in Memphis have been inspired by Justine's, and Justine's itself was given high accolades by Craig Claiborne, who was one of the most famous restaurant critics of the United States; I'd love one day to pick up a copy of Craig Claiborne's Southern Cooking, but that's another matter.
The Justine's book doesn't honestly present a whole lot of recipes, since fully half the book is just about the restaurant, the "memories" part. Fortunately the recipes that are included are nearly exclusively fantastic, with her Creme Spinach having become a Thanksgiving favorite, having become a yearly institution for us. And of course songs have been sung of the virtues of the Crab Justine, which is just unreasonably good. Some of the recipes are just one line, such as for artichoke hearts: "Served with a choice of salad dressing or Hollandaise." BRILLIANT! Later Justine's French dressing is recommended for the salad dressing, which does sound very good.
I do not prefer Justine's Bearnaise, Hollandaise, or French dressing, but each of those can be made from hundreds of recipes (in fact I find the Joy of Cooking versions to be superior, if complicated). I think that where the Justine's cookbook excels is that all of the recipes are sort of deceptively simple (the Southern influence) versions of classic French/Continental dishes.
I'm sure that there are better books, at this point, dedicated to cooking fish, but for me, James Beard's New Fish Cookery is my favorite. Mine is a 1976 printing, ISBN 0-316-15745-7, but I'm sure that the 1994 reprinting that is still in print will be perfectly fine as a replacement. Beard presents the "Canadian Cooking Theory," which I just refer to as "The Method," which dictates that you cook fish for 10 minutes for every inch of fish, no matter what method you are using to cook it, whether it be poaching, broiling, steaming, braising, pan-frying, etc.
Ever since I started cooking fish, I have used this method, and I have had mercifully few failures. There are, of course, caveats for this (for instance, if you pile a lot of garnish on top of the fish it takes longer for the heat to penetrate, so you have to cook it for longer), but as a general rule, it works very well. I do not abide by the modern trend of undercooking fish, since to me undercooked fish is flavorless and slimy, and using "The Method" will ensure that the fish is, actually, cooked. There are a lot of other reasons to cook fish all the way through, bacteria and parasites and others chief among them.
There are many other simple recommendations for improving the consistency, flavor, and preparation for fish, and so the first 20 pages of this book have probably been seen by my eyes more than any other page in the book, save the sauces chapter (pp. 21-38), which is straighforward, simple, and excellent.
The majority of the book is simply how to prepare fish, alphabetically by fish, presenting mostly very solid, traditional preparations. One example is the Lobster au Gratin, which is as delicious as it is simple: Lobster, butter, Cognac, wine, cream, and hollandaise, broiled until bubbly, served over rice.
If you go to a fish market that has some crazy, unusual stuff in the cooler, you just need a copy of this book and you'll immediately know what to do with them. Most fish, prepared well, I consider to be a joy, though I still do occasionally have some failures (such as the excellent cut of Spanish Mackerel I bought last night, that I proceeded to completely ruin under the broiler). I believe that probably "fish theory" has moved beyond James Beard's New Fish Cookery, especially considering the techniques provided from other world cuisines (for instance I refuse to serve crab without having "cured" it first in amazu), but it is very difficult to go wrong with any of the recipes or methods in this book.
There is no reason to buy an Italian cookbook at all if it is not called The Silver Spoon, ISBN 0-7148-4531-0. This is an authentic Italian cookbook, which is essentially the Italian version of Joy of Cooking, having only been translated into English for the first time in 2005. If you want to know how to cook anything Italian, you buy this book. Plain and simple. Additionally, if you have someone breaking into your home and you need to subdue them, all you have to do is drop this book on their heads; it is over 1200 pages long and weighs about 10 pounds.
Besides, it is only 40 bucks. The considerably smaller Japanese Cooking, A Simple Art was 45 dollars, and has probably 1/5 or 1/6 the number of recipes and information that The Silver Spoon contains.
I am certain that, if I had ever been able to afford it, Julia Child's Mastering the Art of French Cooking would be on this list, but since the 2-volume set runs for 90 dollars, I've never had the privilege of cooking out of it. If you're hurting for Christmas present ideas for me, there's a hint :)
But for real, practically everything I know about cooking came first out of the Joy of Cooking. It's joyful.