This text is describing the records that this court (King's County Supreme Court, a court of first instance) has of a certain case. The "type" of case is "CONF Matr Uncontest."

Here's a link from that very court with a FAQ on divorce cases. It is entitled "Matrimonial (Divorces)," so obviously "Matr" in the "Type of case" section means it was a divorce case: http://www.nycourts.gov/courts/2jd/kings/civil/matrimonial.s...

There are two types of divorce: contested (one spouse doesn't want the divorce, or they both do but they disagree about who gets what) and uncontested (both spouses want it, and they agree about who gets what).

Apparently this court files those two types of cases separately, as either "Matr Uncontest" or, presumably, "Matr Contest." According to the link ("Overview" section), uncontested divorces are processed by court clerks. Presumably contested ones are handled by judges.

Also according to that link, divorce cases are confidential. Normally in the US, civil and adult criminal (but not juvenile criminal) case records are entirely public; anyone can access them. In New York, apparently, divorce records are not public; they're confidential. (See question 3 at the link for more on that).

So that would be why CONF is indicated in all-capitals at the very start of the file: to remind court clerks that unlike every other type of civil case, this particular case is confidential so records should not be released to anyone but the parties or their lawyers.

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Note added at 3 days 8 hrs (2019-02-21 04:37:32 GMT)
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PS: In New York state, unlike every other US state, the courts of first instance are called Supreme Courts (X County Supreme Court). Every other state, and the federal government, uses "Supreme Court" to mean the final court of appeal.

New York cases that get appealed proceed like so: (1) filed and decided in X County Supreme Court; (2) appealed to the Supreme Court, Appellate Division; and finally, (3) appealed to the New York Court of Appeals.

1e instance: https://en.wikipedia.org/wiki/New_York_Supreme_Court

1e appel: https://en.wikipedia.org/wiki/New_York_Supreme_Court,_Appell...

Dernier appel: https://en.wikipedia.org/wiki/New_York_Court_of_Appeals

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Note added at 5 days (2019-02-23 14:17:57 GMT)
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@PhB: we don't need to find a trace in the Civil Code or elsewhere in French law, because this is a New York legal term that means a specific thing in New York -- there isn't necessarily an exact equivalent in French. Even American non-lawyers won't be 100% sure exactly what an "uncontested divorce" involves. You just need to translate the words; anyone who cares exactly what their ramifications are in New York divorce litigation would have to look it up, whether the words are in English or in French.