When is longest night?

Depends on where you are...

In the northern hemisphere, the longest night is around Dec. 21, 22.
The further north you go, the longer the night lasts. Near and above
the artic circle, the sun will not appear to rise at all on those
days.
In the southern hemisphere, the longest night is around June 21, 22.
Same principle, the further south you go, the longer the night is.

Or, if you are planning a trip to Venus, be prepared for a nice 243 day evening...

The longest night in the northern hemisphere occurs on the solstice
that happens in December.  Our northern-hemisphere-centric culture
usually refers to this as the "winter solstice", but it is actually
the summer solstice for folks living south of the Earth's equator (the
southern hemisphere's *shortest* night).  Similarly, the southern
hemisphere's longest night and the norther hemisphere's shortest night
occur on the June solstice.

The actual dates of the solstices are not fixed because of the details
of the Earth's rotational and orbital motions, and the nature of our
calendar (which has a built-in ~4-year cycle due to leap years).  A
listing of the dates of the solstices and equinoxes from 1992 to 2020
can be found at the US Naval Observatory's website at
<http://aa.usno.navy.mil/data/docs/EarthSeasons.html>.  Other relevant
pages there include:
<http://aa.usno.navy.mil/faq/docs/longest_day.html>
<http://aa.usno.navy.mil/faq/docs/dark_days.html>
and
<http://aa.usno.navy.mil/faq/docs/equinoxes.html>.

Another good summary can be found at: 
<http://en.wikipedia.org/wiki/Winter_solstice>

Then Joshua spoke to the Lord in the day when the Lord delivered up
the Amorites before the children of Israel, and he said in the sight
of Israel: Sun, stand still over Gibeon; and Moon, in the Valley of
Aijalon. So the sun stood still, and the moon stopped, till the people
had revenge upon their enemies. Is this not written in the book of
Jasher? So the sun stood still in the midst of heaven, and did not
hastened to go down for about a whole day. And there has been no day
like that, before it or after it, that the Lord heeded a voice of a
man; for the Lord fought for Israel.
--Joshua 10:12-14

If the sun was shining especially long on Joshua in Gibeon, then the
sun was not on the other side of the world that whole time.  Thus that
night on the other side of the world "there has been no [night] like
that, before it or after it".

I think this might be the winner.

Hi Jack: One reason we have so many religions is people read extra
inferences into the Bible. It seems impossible to halt the Earth's
rotation for a day without lots of other things happening, so perhaps
this was local and did not affect the length of night on the opposite
side of the Earth. In any case, 47 hour and longer nights occur
throughout most of the Arctic and Antarctic regions during parts of
each fall/winter season.  Neil