First, let’s deal with the pronunciation. The `di’ is pronounced like the word `die’ and the following `es’ is like the word `ease’. The second word, which has the main stress, is pronounced like the word `non’.
`Dies non’ is the shortened form of the Latin expression `dies non juridicus’. `Dies’ means `day’, `non’ means `not’ and `juridicus’ means `judicial’. As you have probably guessed by now, this expression is mostly used in legal contexts to mean a day on which the court doesn’t work; a non-judicial day. Sundays, for example, are non-judicial days.