What does an apostrophe (') mean in English translations of Japanese words?
For example, an online grammar site says, "When you contract te oku to
t'oku…."

Although just a beginning-level student, I have come across this numerous
times, both online and in English-language books. I do not know how to
determine what kana have been replaced by the apostrophe. Or, perhaps it
represents a brief pause in pronunciation, like that made with doubled
consonant sounds?
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question
translation punctuation
Meaning of Apostrophe in Words Translated to English
Googling with the word "transcription," I was able to compile the following
answer:

The apostrophe is used before the romaji letters a, i, u, e, o, y when they
follow n/ん in order to distinguish, for example, kani from kan'i. The
apostrophe makes it clear that ren'yōkei means …んよ… and not …にょ… as in
renyōkei.

The Japanese government’s guidelines for romanization suggest inserting
apostrophes in these situations:

> はねる音を表わす「 **n** 」と次にくる母音字または「 **y** 」とを切り離す必要がある場合には、「 **n** 」の次に「 **’**
> 」を入れる。

The apostrophe is also part of the romanization system used by the U.S.
Library of Congress, a system based on the revised Hepburn system (first laid
out in Kenkyusha's New Japanese-English Dictionary, 3rd ed.):

> Transcribe the apostrophe ( ' ) between syllables when the first syllable
> ends with the letter n and the following syllable begins with the letter a,
> i, u, e, o, or y and when it is necessary to separate romanization.

Examples:

> toshokan'in/図書館員  
>  Nagai Ken'ichi/長井憲一  
>  Shin'etsu/信越  
>  hon'yaku/翻訳  
>  Man'yō/万葉  
>  shin'ainaru/親愛なる  
>  san'okuen/三億円  
>  shin'uchi/真打  
>

From Wikipedia:

> In the revised Hepburn system the apostrophe is used to mark “the separation
> of easily confused phonemes (usually, syllabic n/ん from a following naked
> vowel or semivowel). For example, the name じゅんいちろう, is written with the kana
> characters ju-n-i-chi-ro-u, and romanized as Jun'ichirō in Revised Hepburn.
> Without the apostrophe, it would not be possible to distinguish this correct
> reading from the incorrect ju-ni-chi-ro-u. This system is widely used in
> Japan and among foreign students and academics.”
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answer
The case where the apostrophe is used to avoid ambiguity is already mentioned
in the comments of the question and NattoYum's own answer, but this is not the
case in "t'oku".

As noted by psosuna and others, it means contraction, and the change from "te
oku" to "t'oku" is [音便](https://ja.wikipedia.org/wiki/%E9%9F%B3%E4%BE%BF). I
don't think that is standard in transliterated Japanese however, and I have
seen many grammar sites that introduce this grammar without using the
apostrophe.
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The typical use of an apostrophe in Romanized Japanese is to show that the
original Japanese is N (ん) and not one of the na-ni-nu-ne-no (なにぬねの) series of
letters. Sometimes these can be confused when Romanizing the original
Japanese.
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answer
The original, historical purpose of the apostrophe is to show that something
is left out:

>   * le homme → l'homme (an obligatory contraction in French)
>   * I am → I'm (an optional contraction in English)
>

In other words, it marks a contraction.

It's used that way in most modern Latin orthographies and is not specific to
English. It is not commonly used that way in Latin transcriptions
("romanization") of Japanese, but if an author is specifically discussing
contractions in a linguistic context it should be clear what it means:

>   * _-te oku_ → _-t'oku_ (an optional contraction in Japanese)
>

The /e/ sound is elided and the apostrophe (in this case) indicates that
contraction. The transcription _-te oku_ represents the same sounds as ておく,
and _-t'oku_ represents the same sounds as とく. The only difference is that the
Latin transcriptions use a dash to show an affix boundary and an apostrophe to
mark elision.

No kana has been replaced with an apostrophe, as you write; the kana て
represents a sequence of two phonemes, only one of which has been elided.

Although contractions are not marked in most transcriptions of Japanese, this
convention has been used by linguists such as Samuel Martin, and it should be
readily understandable in the proper context - if someone writes "contract te
oku to t'oku", the only sound that could have possibly been left out is /e/.
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answer