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.
I'd rather quote Knuth himself (The TeXbook, page 1):
English words like "technology" stem from a Greek root beginning with
the letters τεχ...; and this same Greek word means art as well as
technology. Hence the name TeX, which is an uppercase form of τεχ.
Insiders pronounce the χ of TeX as a Greek chi, not as an "x", so that
TeX rhymes with the word blecchhh. It’s the "ch" sound in Scottish
words like loch or German words like ach; it’s a Spanish "j" and a
Russian "kh". When you say it correctly to your computer, the terminal
may become slightly moist.
After reading some comments to this answer, I think the following quote is also appropriate (it appear just one paragraph below the previous quote):
On the other hand, it’s important to notice another thing about TeX's
name: The "E" is out of kilter. This displaced "E" is a reminder that
TeX is about typesetting, and it distinguishes TeX from other system
names. In fact, TEX (pronounced tecks) is the admirable Text EXecutive
processor developed by Honeywell Information Systems. Since these two
system names are pronounced quite differently, they should also be
spelled differently. The correct way to refer to TeX in a computer
file, or when using some other medium that doesn’t allow lowering of
the "E", is to type "TeX". Then there will be no confusion with
similar names, and people will be primed to pronounce everything
properly.
Best Answer
actually, there was setting of math with computers before tex. very specialized, and usually quite expensive. in what follows, "symbol" is used to mean either a letter of some alphabet, a digit, or some other shape.
there are several phases in the history of math composition:
hand-set type -- anything before the late 1800s.
monotype -- several patents from 1885-1890 covered this system and its basic operation for ordinary text. the wikipedia article covers this pretty well. what it doesn't mention is the "4-line system" that was required for math. this is described in daniel rhatigan's m.a. thesis. this is the system used to compose the first editions of the first two volumes of knuth's "the art of computer programming", and it was the switch away from monotype that is the reason that tex exists today.
"typewriter" composition. actually, this is an oversimplification. the "direct type" methods were of two distinct kinds. both were best used on machines with a carbon rather than an inked ribbon.
traditional one-symbol-per-key typewriters. an "ordinary" typewriter has keys linked to type bars containing usually two symbols, with a "shift" key to change from lower to uppercase. any symbol not on one of these type bars had to be added by another method, either by hand or by the use of a widget called a "typit", a plastic device with a raised symbol at one end that could be stuck into a guide mounted on the typewriter where a key would impact the ribbon, and any key could be struck to impact the typit and transfer the image of that symbol to the ribbon and paper. typits were used on machines with either proportional or fixed spacing.
a machine with interchangeable "fonts". i know of two kinds used for direct manual input.
the "varityper", had a revolving drum onto which two semicylindrical fonts could be mounted, allowing both latin and greek (for example) to be available at the same time. switching one font for another was reasonably fast, and the use of proportional spacing resulted in an appearance that was more refined than possible with a monospace typewriter, but not nearly the quality of real composition.
the ibm selectric and "clones" used a "golf ball" mechanism. these type units were easy to switch, although only one "font" (containing the same number of symbols as keys on the keyboard, times however many shift keys were provided) could be in use at one time.
photocomposition was the next advance. i know of one direct-input photocomposer used for math -- the photon 200. a keyboard with several automobile-like shift levers controlled the input. images of symbols were transparent shapes on a glass disk otherwise coated with black, like a photographic negative. for "pi" symbols not on the disk, there was a "whirligig" with eight arms, onto each of which could be mounted a photographic negative of one symbol; jobs had to be carefully edited beforehand to make sure that no more than the eight available pi symbols would be needed, since changing the pi set in the middle of a job could run the risk of exposing the output medium, which would mean starting over. there was sufficient computer power in this machine to calculate line length and apply justification (in text), but for the most part, such decisions for math were manual, and required a highly skilled operator.
for more automated photocomposition, input was keyed into some recording medium -- punched cards, paper tape, ocr forms, floppy disk, ... -- and input to the typesetting machine for output to photographic paper. since photographic paper was messy and expensive, there were sometimes preliminary proof runs in which the input code was printed out on greenbar; not ideal, and accurate really only for text, but often less expensive than other alternatives. (and patching paper tape was definitely an "interesting" undertaking.)
output devices varied in how images were transferred to the photographic medium.
the photon 713 had a cylindrical drum into which were mounted 35mm film strips, each with about the equivalent of what would fit on a selectric golf ball. there were 16 of these fonts, so the available collection included roman, italic, bold, greek, and a reasonably comprehensive collection of math symbols. the drum rotated at high speed, and the images were set by a carefully timed strobe while the disk assembly was advanced across the width of the paper by an absolutely gorgeous worm gear.
crt output was another possibility. these devices were originally devised for use by newspapers, where speed was essential, and cost secondary. the imaging process usually involved moving a crt beam along a predetermined path, turning it on and off as appropriate. the symbols in fonts were designed by hand, which involved defining the paths for the beam. only certain directions were possible -- vertical, horizontal, and at 45 degrees, but one of these was not permitted (i've forgotten which), so it was necessary to make sure that a symbol could be completed in a single, continuous path, avoiding the proscribed direction.
there were a few small companies that developed math composition software before tex. one was science typographers, whose program was used in-house at ams for about 20 years; a discussion of the impact of tex on the suppliers of commercial math composition software appeared in tugboat in 1983. (on rereading what i wrote then, i find that i haven't changed my mind about what makes tex a good choice for composition in general, and the most compelling argument against it has dissipated with advances in hardware technology.)
one other essentially non-commercial typesetting system deserves mention -- troff -- which, with its companion
eqnwas created at bell labs and available on unix systems from the early 1970s, but which originally produced typeset output only on the cat typesetter, and was freely available only to unix sites.this review has gone on long enough. if anything significant is missing, or any errors are noted, please let me know, and i'll try to remedy the situation.