The most important single capacity human beings have is the capacity to reason logically  to know that from given premisses, whether believed or not, follows something with necessity if these premisses are true.
There are several reasons why this is the most important single capacity human beings have:

Some reasons why logic is of fundamental human(e) importance


It is at the foundations of all human inferences, and therewith of human survival

It is at the foundations of peaceful, rational and reasonable social change and all argumentation

Together with language, logic is what makes human beings really different from other animals: only human beings can seek peaceful agreement on what may be the case and what may be done by rational discussion

Without mathematical logic, computers are hard to think of, design, built or maintain

Reviewed March 26 2012:
1. There are quite a few sites dedicated to logic in some sense. Most are maintained by academics who cater mostly or only to a few fellowacademics in the same specialism. And while there is nothing wrong with that, it would be pleasant if I could direct you to a site where you can learn logic at most levels and enjoy the experience. Unfortunately, sofar I have neither found that site nor made it myself. (Maybe in the future, health permitting.)
Those who want to link to some interesting sites with accessible, amusing, interesting and civilizing ideas, formulations, games, puzzles relating to (mathematical) logic and related subjects are advised to check the following site and the ones after it:
Roger Bishop Jones 
Roger Bishop Jones's Factasia

"Factasia is a philosophical fantasy about the future of society and the future of technology." and indeed it is, and it contains a lot of logic, philosophy, and many bookreferences and links. This is very well done, but to delve deeper in the logic on the site you need more than is on that site. The bookreferences to do so are there  and I recommend that you download Mr Jones' site in the zipped version of Factasia he provides for that purpose, if his site is even a little to your taste, for it is large and wellorganized, and far easier and cheaper to access once it is on your hard disk. Also, it'll probably teach you a lot directly or indirectly if you are in any way seriously interested in philosophy, logic, computers or mechanical proofs.
At age 15 or so I probably would have committed murder to be able to read this material. Now you can do so for free on the internet.
2. If you really want to understand both the beauty and the use of mathematics and logic you have to see it applied to all manner of problems. Here are four pages that contain a great amount of links to show just this
Frank Potter's Science Gems  Mathematics
 Physical science
 Physical science ii
 Physical science iiii

3. Next, there are some bookreferences to explain what I mean by "logic". For the moment I list only authors and titles, and do not know what is in print. All titles except the last recent ones should be available in any decent university library. Also, when I reviewed this list in March 2012, it seemed nearly all titles I mention are available second hand on the internet, while some are still in print.
Classic expositions

Bertrand Russell

Introduction to Mathematical Philosophy

Alfred Tarski

Introduction to Mathematical Logic

Hasenjaeger

Introduction to the Basic Concepts and Problems of Modern Logic. 
Paul Halmos

Naïve Set Theory

Bochenski

Formale Logik

Good introductions

Evert Beth

Foundations of Mathematics

Van der Waerden

Algebra

Joseph Shoenfield

Mathematical Logic

Herbert Enderton 
A mathematical introduction to logic 
Geoffry Hunter 
Metalogic 
Marvin Minsky

Computation: Finite and infinite machines

Good recent books

F.A. Muller

Structures for everyone

Barwise & Moss

Vicious Circles (and more: see below)

George Boolos

Logic, Logic and Logic (and more)

Raymond Smullyan 
First Order Logic (and more) 
Here a few comments on these titles  preceded by the general comment that one fundamental criterion to list them is their clear styles of writing:
Bertrand Russell

 Introduction to Mathematical Philosophy

This is an exposition of the intuitions and mathematics that went into Russell's "The Principles of Mathematics" and Whitehead & Russell's "Principia Mathematica", mostly without symbolism and accessible to anyone with a clear mind.
Alfred Tarski

 Introduction to Mathematical Logic

In several ways the best introduction to the subject, especially because it is nonpretentious and clear about fundamentals.
However, neither Russell's or Tarski's abovementioned texts go far mathematically (and were not meant to be). One of the best introductions to the more mathematical side of logic is
Gisbert Hasenjaeger

 Introduction to the Basic Concepts and Problems of Modern Logic. 
This is especially good because he really goes into the intuitions behind the mathematics, and also contains good expositions of stuff usually not found in other introductions, while being formally both rather clear and precise. (It may be that the English title I found and quote is not quite the same as the German text I read.)
Paul Halmos

 Naïve Set Theory

Nearly all mathematics these days at least uses the notation of settheory and presumes an understanding of its foundations. Halmos wrote a very clear introduction, and also wrote several interesting books that treat logic as a part of algebra: See his Algebraic Logic for polyadic algebra.
J.M. Bocheński

 Formale Logik

There is much more to logic than modern mathematical logic. This is the best history of logic in Western thought I've seen. (There also is a fine Indian tradition, impressively summarized by  I believe at present six  volumes of Link to: Karl H. Potter (I have no idea whether he is family of Frank Potter above))
Evert Beth

 Foundations of Mathematics

Very wide ranging survey of the subject by a great Dutch mathematical logician. Subjectwise it is a bit out of date, but stylistically and conceptually it is not.
Something similar holds for the next book, that sheds lots of light on mathematical logic from a mathematical point of view
Van der Waerden

 Algebra

This book  in fact originally 2 volumes in German  is close in spirit (but much older) than the expositions in Muller's and Halmos's books mentioned below. It also is concerned with Algebra in the mathematical sense, which covers a lot: logic, groups, operators, matrices, fields etc.
Joseph Shoenfield

 Mathematical Logic

There are many mathematical expositions of mathematical logic. Shoenfield I found the clearest. It also covers a lot of material in a fairly small scope.
Herbert Enderton

 A mathematical introduction to logic

In some ways the clearest, simplest and most thorough exposition. Somewhat less fastpaced than Shoenfield.
Geoffrey Hunter

 Metalogic

Another fine basic exposition, especially fit for people who did not study mathematics but who want a mathematically adequate and clear exposition.
Marvin Minsky

 Finite and infinite machines

This is an excellent very readable introduction to the mathematical ideas involved in computing (for which you don't need much mathematics: a clear mind is all that is necessary).
None of the books I've mentioned sofar has been recently published (or if it was, like Tarski 's text I mentioned, it is a reprint). The next few books are recent:
F.A. Muller

 Structures for everyone

This is the recent doctoral thesis of a Dutch mathematical physicist. It covers a lot of material, including Quantum Mechanics, but has the great advantage of being very clear about what theories are supposed to be. Muller also delves quite deep into the foundations of set theory and of category theory.
In general terms, he expounds a version of Bourbaki's structuralist approach to mathematics based on a version of Ackermann's theory of sets and classes, using Sneed's, Suppes', and Stegmuller's structuralist account of what scientific theories are. As the reader may have gleaned, the general point of view is: Everything  absolutely everything  is a structure of some kind.
Barwise & Moss
Barwise & Etchemendy

 Language, Proof and Logic
 Vicious Circles
 The Liar: An Essay in Truth and Circularity

The first of these is an introduction to logic that may have been the first such book to include computer exercises. I read it long ago, liked it, and gave it away, so can't say much more about it, except that I liked it.
The second is an exposition of paradoxes and vicious circles. It contains a lot of good clear explanations of recent thinking in mathematical logic in fields related to this subject including computer programming and theories of truth.
In general terms, the authors sketch solutions (or approaches to solutions) based on the idea to give up one of the standard axioms of set theory, the Axiom of Foundation, that excludes the existence of sets that are members of themselves.
This is also interesting for psychology and philosophy of mind, since so many issues in these fields involve some kind of selfreference (such as the one that allows the reader to understand that in this sentence I am saying something about this sentence and myself using the term "I").
The third is a treatment of the paradox of the liar that involves a distinction of two kinds of negation. I am partial to that  kind of  distinction (having thought of it myself in 1975, although I discovered later others did so much earlier) and an interesting attempted solution of a very tricky problem. For more attempted solutions (along various lines) see Recent Essays on Truth and the Liar Paradox, ed. R.L. Martin.
George Boolos

 Logic, Logic and Logic
 Computability and Logic (with R. Jeffrey)

The first is a recent collection of essays by Boolos. It consists of articles in three loose groups (whence the thrice repeated "Logic"), namely about the foundations of set theory (which is the foundation of mathematics, which is the foundation of everything else  briefly), about Frege's logical theories, and about various logical subjects, notably Gödel's theorems and things impossible or impractical in first order logic.
This is also interesting for psychology and philosophy of mind, especially because Boolos discusses higherorder logic (involved in such statements as: "There are some relations and properties thereof I can think of you cannot think of  as shown by this sentence, which you, dear reader, cannot possibly believe to be true") and gives examples of formulas computers can't compute.
The second is an introduction to logic by Boolos and Jeffrey that is good and clear and includes an exposition of Gödel's Theorems and computable functions. I owe the first edition; there have been later ones (with some corrections of the first and some new material).
Raymond Smullyan

 FirstOrder Logic
 Diagonalization and SelfReference  To Mock a Mockingbird  Forever Undecided  Satan, Cantor and Infinity

I really should have included Smullyan but forgot to do so in the original edition of this internet page (that's quite popular, I found to my pleasant surprise). I can recommend all of his books (I think: I have read most of them, and what I have read was uniformly excellent, very readable, and very clear) also those which are not mathematical or logical. (So, in mitigation I have provided links to surveys of these books on Wikipedia.)
In fact, Smullyan published quite a lot of books in three fields, mostly: Mathematical Logic, Logic Puzzles and Philosophy, though there tends to be a substantial overlap that consists mostly of logic.
FirstOrder Logic is a very fine, very clear exposition of propositional and first order logic including metatheorems (theorems about what systems of logic can and cannot prove, and/or about consistency of and provability in logical systems). It is based on a particularly clear version of Beth's Semantic Tableaux, and includes what is probably the clearest exposition of the logic of quantifiers.
Diagonalization and SelfReference: Smullyan got wellknown as a mathematical logician with his Theory of Formal Systems, another very clear introduction to the subject of what formal systems are, precisely, and with work on Gödel's Theorem, summarized in his Gödel's Incompleteness Theorems. The book I listed contains versions of most of the material of these books, and also of another one Recursion Theory for Metamathematics, and is probably Smullyan's main work in mathematical logic. It is unlikely you'll find clearer expositions of the subject, but it should also be said these are genuinely difficult subjects.
Then again, for those who want to understand Gödel's Theorems and have a good time, there is (among others):
Forever Undecided which introduces these theorems and the ideas behind them in the form of a series of logic puzzles, that also introduce standard logic. This is listed as one of Smullyan's logic puzzle books on Wikipedia, which is right in a way  if one realizes Smullyan is the Lewis Carroll of the 20th Century and all his many puzzle books not only contain very clever, very amusing and often quite challenging logic puzzles (with clear solutions!), but in fact are all also introductions to logic.
Another example of this is:
Satan, Cantor and Infinity: It does consist of logic puzzles, but it is in fact also an introduction to standard logic and set theory, that also is one of the most enlightening and amusing introductions to these subjects (mostly without formalism, but nevertheless quite precise and clear).
Finally, for those really interested in logic:
To Mock a Mockingbird: This is a book of logic puzzles that also is an introduction to combinatory logic, which is a foundation for logic and mathematics thought up, created or developed by Schönfinkel and Curry and later by others, that manages to derive logic and mathematics from a the logical combinators Kxy = x and Sxyz = xz(yz). It is quite amazing if you believed Russell and Whitehead's Principia Mathematica or Zermelo's Set Theory are what the foundations of lohic and mathematics should look like or presume. (See also: Lambda calculus. For an exposition of the relation between these and other subjects, see Peter Selinger's "Lecture Notes on the Lambda Calculus".)