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".)