1
It is cold in the night.
It is cold at night.
Use at night to talk about nights in general: ‘It is cold at night.’
For other parts of the day, use in: ‘in the morning’, ‘in the afternoon’, ‘in the evening’.
Use in the night when something happens during one particular night: ‘I woke up twice in the night.’
2
Yesterday night we had a few beers.
Last night we had a few beers.
Use last night, not yesterday night: ‘I didn’t sleep well last night.’
However, we say ‘yesterday morning’, ‘yesterday afternoon’ and ‘yesterday evening’.
Check our grammar lesson on prepositions of time.