This is the 9839th most frequent Russian word.
августовского
"Августовского" means "of August" or "August's" in English, indicating possession or relation to the month of August.
The word "августовского" is used as an adjective modifying "решения" (decision) to specify that the decision relates to August.
Обсуждение августовского решения заняло весь день.
The discussion of the August decision took all day.
Here, "августовским" is an adjective modifying "вечером" (evening), indicating that the described evening occurred in August.
Мы наслаждались теплым августовским вечером.
We enjoyed the warm August evening.
In this sentence, "августовских" is used as an adjective describing "цветов" (flowers) to denote their association with August.