What is the proper use of @ (i.e., backslash-at)?

The macro simply says

spacefactor 1000

Under nonfrenchspacing, capital letters set the space factor to 999 and, by rule, the space factor never jumps from a value less than 1000 to a value greater than 1000. On the other hand, a comma sets the space factor to 1250, the period to 3000 and so on for other punctuation signs.

So with JS,, the space factor after the comma is 1000

J (999) S (999) , (1250->1000)

while with JS,@ one has

J (999) S (999) , (1250->1000) @ (1000)

and with JS@, we get

J (999) S (999) @ (1000) , (1250)

so that the right syntax is the latter.

Let's consider e.g., e.g.@ and e.g@.:

e (1000) . (3000) g (1000) . (3000) , (1250)
e (1000) . (3000) g (1000) . (3000) @ (1000)
e (1000) . (3000) g (1000) @ (1000) . (3000)

so the right syntax is the middle one. However, a comma just after the second period would make @ unnecessary.

You should type your phrase as

Languages like JS@, HTML@, etc.@ were not used by King Henry III@.

otherwise the space after the commas would not expand (factor 1.25) and the space after etc. would (factor 3000), which is not desired.

The simplest thing is to go frenchspacing and forget about this.

This is an old question and this is not so much an answer, but an overly long comment.

The @ before the commas (as is shown in egreg's excellent answer) is not redundant. TeX's treatment of spacefactor is straight-forward, but obscure.

The TeXbook and TeX by Topic have all the details—and most of this is taken from TeX by Topic—but essentially the following happens. When TeX inserts a space between words, the amount of space inserted depends on several factors. The key factors are the current font parameters 2, 3, 4, and 7, along with spaceskip, xspaceskip, and spacefactor. spaceskip and xspaceskip are used to override the font parameters. As they're rarely used, let's ignore them here.

Normally, when a space is inserted, it has a natural width (font parameter 2), a stretch component (font parameter 3) and a shrink component (font parameter 4). The spacefactor modifies this: The stretch component is multiplied by spacefactor/1000 whereas the shrink component is divided by spacefactor/1000.

When spacefactor is at least 2000, then TeX increases the natural width of the space by font parameter 7. This is done to have wider spaces between sentences.

Each character added to the a line has an associated space factor code, sfcode. The spacefactor is set after each character as follows. Let n be the sfcode for the character.

• If the n = 0, spacefactor is unchanged. (This enables punctuation like parentheses and brackets to not affect the spacefactor.)
• If n > 1000 and spacefactor < 1000, then the spacefactor is set to 1000. (This slightly confusing behavior enables D. Knuth to use a normal space.)
• Otherwise, spacefactor is set to n.

Finally, the spacefactor is set to 1000 at the beginning of paragraphs (including after indent or noindent), as well as after vrule, an accent, or a box in horizontal mode.

So, with that in mind, let's consider some consequences of this using Computer Modern 10pt roman fonts.

1. Spaces have an unmodified (by spacefactor) glue of "3.33333pt plus 1.66666pt minus 1.11111pt" which is to say their natural width is 3.33333pt but can stretch by an additional 1.66666pt (without penalty) and shrink by 1.11111pt.
2. Spaces following lower case letters (which have sfcode 1000) have exactly the glue listed above.
3. Spaces following upper case letters (which have sfcode 999) have glue 3.33333pt plus 1.66498pt minus 1.11221pt. This is imperceptibly different from spaces following lower case letters.
4. Spaces following commas (which have sfcode 1250) which themselves follow a lower case letter (e.g., the space in foo, bar) have glue 3.33333pt plus 2.08331pt minus 0.88889pt.
5. Spaces following periods (which have sfcode 3000) which themselves follow a lower case letter (e.g., the space in foo. bar) have glue 4.44444pt plus 4.99997pt minus 0.37036pt. Note that the natural width has been augmented by font parameter 7 because sfcode >= 2000.
6. Spaces following periods (or commas) which themselves follow an upper case letter have the same glue as in 1.

Let's now consider the three short paragraphs in the following LaTeX document.

documentclass{article}
tracingoutput=1
showboxbreadth=1000
showboxdepth=1000

begin{document}
Languages like JS, HTML, etc. were not used by King Henry III. Next.

Languages like JS, HTML, etc. were not used by King Henry III@. Next.

Languages like JS@, HTML@, etc. were not used by King Henry III@. Next.
end{document}

Note that I've instructed TeX to write out what each page of output looks like to the log file. This output is pretty verbose, but you can see each box, glue, and character. For example, the first paragraph consists of this.

...hbox(6.94444+1.94444)x345.0, glue set 16.08281fil
....hbox(0.0+0.0)x15.0
....OT1/cmr/m/n/10 L
....OT1/cmr/m/n/10 a
....OT1/cmr/m/n/10 n
....OT1/cmr/m/n/10 g
....OT1/cmr/m/n/10 u
....OT1/cmr/m/n/10 a
....OT1/cmr/m/n/10 g
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 s
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 l
....OT1/cmr/m/n/10 i
....OT1/cmr/m/n/10 k
....kern-0.27779
....OT1/cmr/m/n/10 e
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 J
....OT1/cmr/m/n/10 S
....OT1/cmr/m/n/10 ,
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 H
....OT1/cmr/m/n/10 T
....OT1/cmr/m/n/10 M
....OT1/cmr/m/n/10 L
....OT1/cmr/m/n/10 ,
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 t
....OT1/cmr/m/n/10 c
....OT1/cmr/m/n/10 .
....glue 4.44444 plus 4.99997 minus 0.37036
....OT1/cmr/m/n/10 w
....kern-0.27779
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 r
....OT1/cmr/m/n/10 e
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 n
....OT1/cmr/m/n/10 o
....OT1/cmr/m/n/10 t
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 u
....OT1/cmr/m/n/10 s
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 d
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 b
....kern-0.27779
....OT1/cmr/m/n/10 y
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 K
....OT1/cmr/m/n/10 i
....OT1/cmr/m/n/10 n
....OT1/cmr/m/n/10 g
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 H
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 n
....OT1/cmr/m/n/10 r
....OT1/cmr/m/n/10 y
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 I
....kern0.27779
....OT1/cmr/m/n/10 I
....kern0.27779
....OT1/cmr/m/n/10 I
....OT1/cmr/m/n/10 .
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 N
....OT1/cmr/m/n/10 e
....OT1/cmr/m/n/10 x
....OT1/cmr/m/n/10 t
....OT1/cmr/m/n/10 .
....penalty 10000
....glue(parfillskip) 0.0 plus 1.0fil
....glue(rightskip) 0.0

There's a lot of interesting stuff here which I'm going to skip over to focus on the key points.

• The glue set after JS, and HTML, are normal spaces: glue 3.33333 plus 1.66666 minus 1.11111
• The glue set after etc. is an inter-sentence space: glue 4.44444 plus 4.99997 minus 0.37036
• The glue set after III. is a normal space.

This is precisely what we'd expect by following the rules given above.

The minimum one should do to fix these problems is the second paragraph:

• The glue set after JS, and HTML, are still normal spaces.
• The glue set after etc. is a normal space.
• The glue set after III@. is an inter-sentence space.

The correct thing to do is what egreg said and given in the third paragraph:

• The glue set after JS@, and HTML@, are glue 3.33333 plus 2.08331 minus 0.88889.
• The glue set after etc. is a normal space.
• The glue set after III@. is an inter-sentence space.

What does this matter? Imagine you had written Ruby, JS, and HTML@. instead. We might expect (and would certainly want) the space following Ruby, and the space following JS, to be the same. Unfortunately, following the rules detailed above, this doesn't happen.

...hbox(6.94444+1.94444)x345.0, glue set 232.08315fil
....hbox(0.0+0.0)x15.0
....OT1/cmr/m/n/10 R
....OT1/cmr/m/n/10 u
....OT1/cmr/m/n/10 b
....kern-0.27779
....OT1/cmr/m/n/10 y
....kern-0.83334
....OT1/cmr/m/n/10 ,
....glue 3.33333 plus 2.08331 minus 0.88889
....OT1/cmr/m/n/10 H
....OT1/cmr/m/n/10 T
....OT1/cmr/m/n/10 M
....OT1/cmr/m/n/10 L
....OT1/cmr/m/n/10 ,
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 a
....OT1/cmr/m/n/10 n
....OT1/cmr/m/n/10 d
....glue 3.33333 plus 1.66666 minus 1.11111
....OT1/cmr/m/n/10 J
....OT1/cmr/m/n/10 S
....OT1/cmr/m/n/10 .
....penalty 10000
....glue(parfillskip) 0.0 plus 1.0fil
....glue(rightskip) 0.0

In this format, it's readily apparent that the two spaces after the commas have different stretch and shrink components. Writing, Ruby, JS@, and HTML@. has the correct behavior.

egreg writes, "The simplest thing is to go frenchspacing and forget about this." It's certainly true that that is the simplest thing to do, but unless it's common to typeset all spaces with the same base width in your language/journal/whatever, I recommend not doing that.

For novices, a simple mnemonic rule is: @ is an invisible (zero-width) lowercase letter.

LaTeX considers a sequence <lowercase letter> <punctuation sign> <space> as end of a phrase and adds extra space for readability: in That's all. Thanks, LaTeX adds more space after all.. Otherwise it doesn't add extra space after the punctuation sign: in A. Einstein, the period does not end a sentence because the preceding letter is uppercase; in pi = 3.14, the period does not end a sentence because the next character is not a space.

However, this rule sometimes fails:

(1) Apple Inc. sells phones

here, LaTeX thinks the period ends a sentence and wrongly adds an extra space.

(2) My school scores were all A. Einstein would be proud of me!

here, LaTeX does not realize that the period ends a sentence and does not add the expected space.

You can correct this using an invisible lowercase letter @:

(1) Apple Inc.@ sells phones 

here, the period does not end a sentence because it is no longer followed by a space.

(2) My school scores were all A@. Einstein would be proud of me!

here, the period does end a sentence because it is preceded by the "lowercase letter" @.

This applies not only to a period but to commas, etc.: Score a, score b, score c. is OK, but with uppercase the correct use is Score A@, score B@, score C@. for LaTeX to realize that each comma ends a phrase (since it goes after a lowercase "letter" @).

(This simplistic explanation is not technically correct; see technical details in other answers.)