Let's say that today is Monday the 24th.
Last night is the night of Sunday the 23rd.
Tonight is the night of today (Monday the 24th).
Tomorrow night is the night of Tuesday the 25th.
I've never heard anyone say 'this night' or 'next night' like that before, but logically they mean 'tonight' and 'tomorrow night' respectively.
'This night' could be used just to talk about the night in general, just like you'd say 'this house' or 'this person', instead of making a statement of time (ex. This night is very nice).