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:
"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
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:
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.
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
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.)
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.
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))
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
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.
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.
In some ways the clearest, simplest and most thorough
exposition. Somewhat less fastpaced than Shoenfield.
Another fine basic exposition, especially fit for people
who did not study mathematics but who want a mathematically adequate
and clear exposition.
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:
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).
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".)