Ennui has two meanings in French, one of which is boredom. The other is annoyance.
When meant as boredom, ennui is prefixed by the definite article, le (the,
masculine form). As ennui begins with a vowel sound, the article is
shortened to l' , making pronunciation smoother. The result is l'ennui,
pronounced lon-WEE.
In the case of annoyance, ennui is prefixed by the indefinite article, un (a,
masculine form). There is no written shortening of the article in this case,
but emphasis is put on the N of un, such as that it almost becomes part
of the first syllable of ennui. This is also to make pronunciation smoother
in the case of a vowel sound. The result is un ennui, pronounced uhnon-WEE.
If ennui is used with a demonstrative pronoun, for example "Quel
ennui!", meaning "What/such annoyance/boredom!"),
it can have either meaning, which can lead to confusion and a loss of meaning
in textual translation. Verbally, the meaning is generally indicated by the
speaker's intonation. A whiny or desperate voice will likely mean boredom,
while a serious or angry voice will tend to mean annoyance.
Ennui also has a related verb: ennuyer (on-WEE-ey), which can
also have either meaning.
The verb is conjugated as follows:
First person singular, j'ennuie (jhon-WEE)
Second person singular, tu ennuies (too-on-WEE)
Third person singular masculine, il ennuie (eel-on-WEE)
Third person singular feminine, elle ennuie (el-on-WEE)
First person plural, nous ennuyons (noozon-WEE-yon. As seen with the
indefinite article nous, meaning we, almost becomes part of the
first syllable of the next word in order to make pronunciation smoother.)
Second person plural or singular/plural politeness form, vous ennuyez (voozon-WEE-ey)
Third person plural masculine, ils ennuient (eelzon-WEE)
Third person plural feminine, elles ennuient (elzon-WEE)
Note that certain French sounds do not exist in the English language. As such,
the pronunciation is approximated using English sounds without resorting to
the International Phonetic Alphabet.
From my Swiss-French point of view, there is no je-ne-sais-quoi about
it other than the general fanciness that surrounds the French language. I feel
it has no relation to Weltschmerz, a German word for worldweariness, though
not unlike in English, one can make it mean worldweariness by saying something
like "Le monde m'ennuie.", "The world bores me." | 
