This is the 5205th most frequent Russian word.
новогодние
Relating to New Year.
Here, 'новогодними' modifies 'гирляндами' and is in the instrumental case, describing the decorations associated with the New Year.
Мы украсили дом новогодними гирляндами.
We decorated the house with New Year's garlands.
In this sentence, 'новогодняя' describes 'ёлка' and agrees with it in gender, case, and number, indicating the festivity related to the New Year.
На площади стояла огромная новогодняя ёлка.
In the square stood a huge New Year's tree.
Here, 'новогодний' describes 'ужин' in the masculine singular nominative case, referring to the special occasion of the New Year dinner.