You should use the environments from amsmath. In practice, equation and align are all you usually need.
If you have a single equation, use equation. (Or equation* if you don't want it numbered. Most of the other environments below also have similar * variants.)
If you have a single equation spanning multiple lines, you can either use multline, or use split (inside equation) to have the parts aligned.
If you have multiple equations and you want them to be aligned, use align (or align*).
If you simply want to typeset multiple equations independently (with no alignment), use gather.
There are also flalign and alignat, for some special cases. See the Short Math Guide for LaTeX or texdoc amsldoc (PDF) for more documentation on these environments.
\[ simply says "set the following in a math display", like plain TeX 's $$ (which you should not use), and is equivalent to displaymath. You can use it if you want an unnumbered equation and are too lazy to type (not good practice, semantically speaking), or, I guess, when you're simply "displaying" some long bit of mathematics that isn't an actual equation. And never use eqnarray.
I think the comments above make as good an answer as any would so I've taken the liberty of assembling them into a single CW answer.
The TeX lion was designed for the illustrations of the first TeXBook. Duane Bibby is the artist. He describes the genesis of the idea in a 2006 interview this way:
Q: How did the lion motif come about?
A: During that first meeting, Don [Knuth] showed me some writing he had
done that had been published in Mad magazine. That not only greatly
impressed me, because I had a fair stack of rejection slips from them,
but helped in thinking about perhaps using a character to bind things
together — somewhat similar to a job I'd just completed with Eleanor
Mennick, the design director at Fearon. Don liked the idea, but we
didn't have a clear vision at that time of exactly what sort of
character. The obvious was, we thought, a kind of computer guy, of
which I did some exploratory sketching around. But, as my wife
Jeanette Ahlgren noted as I doodled in our cabin in the trees, also
kind of boring. I'd also had a chance to read the manuscript and found
the tone light and engaging even though I knew nothing of computer
software. I think that influenced the approach too.
Various animals came to mind and pad, but a classic lion finally began
to pop to life. A possible source of the lion idea was a very large
Maine Coon cat — a rather large breed of house cat — that was
wandering around. It had been abandoned, was looking for a new home,
and was giving us new arrivals the look over, trying to decide if he
would adopt us. He later did. ...
I tried the lion sketches on Don, which he liked right off, and we
then began working out each chapter idea which further defined the
character. Later when it came to Metafont, Don felt the lion needed a
mate and so that made it easy.
The LaTeX Book and The Metafont Book were also illustrated with Bibby's lions, which further enforced the brand.
The TeX Talk logo designed by Pablo is an homage to Bibby. TeX StackExchange users would love to have the lion further incorporated into site graphics, but the money required to hire Bibby has not surfaced. See
CTAN has a lion illustration that is liberally licensed; Jim Hefferon owns it and requests that proper attribution (to the artist) be made. In fact we used it for our captcha screens. (The linked page in turn links to several scanned versions that are available for appropriate uses.)
Some more history: At the "coming out party" for the first hardbound set of Computers & Typesetting, at the Computer Museum in Boston, 21 May 1986, Don Knuth concluded his remarks with a comment on "why a lion?" (p.98). The introductory remarks by Knuth's editor, Peter Gordon, provide more general background about the occasion.
Best Answer
I vaguely remember reading that Donald E. Knuth has frozen the codebase except for actual errors in function that cannot be worked around reliably. His versioning system is a key to what his thoughts on this matter were, his popular projects have version numbers that approach (but can never reach) various trancendental numbers.
Then as mentioned there is no point in rocking a seaworthy boat.
Also we should try and get in Donald's head where he offered bounties for finding errors in his books (of which he paid out some $2.56 cheques of which most were framed and not cashed). He was a pragmatic perfectionist. Do what need to be done to get the job done but no more, the version numbers would keep getting longer if the code was tinkered with endlessly which will discourage random tinkering.
TeX was just Donald's way of getting presentable typesetting for his own books that could no longer be served by Hot metal typesetting and hand tuned equations. We should be thankful that he has had a lot of foresight many of us would have run past blindly.