This is the 13787th most frequent Russian word.
новогодняя
"New Year's" or "related to New Year."
Here, 'новогоднюю' is an adjective modifying 'ёлку', agreeing in gender, number, and case with the noun, depicting an object associated with New Year.
Мы купили красивую новогоднюю ёлку.
We bought a beautiful New Year's tree.
In this sentence, 'Новогодняя' is an adjective describing 'ночь', in the nominative case, referring to a time connected to the New Year.
Новогодняя ночь была очень спокойной.
The New Year's night was very calm.
'Новогодние' is the adjective here agreeing with 'украшения', in the plural form and illustrating items related to the New Year.