This is the 12672nd most frequent Russian word.
коричневого
"Коричневого" means "brown" (genitive or accusative singular masculine/neuter, or genitive plural).
In this sentence, 'коричневого' functions as an adjective describing the color of the shoes ('ботинок'). It agrees in case, number, and gender with the noun it modifies.
Я купил новую пару коричневого ботинок.
I bought a new pair of brown shoes.
Here, 'коричневого' refers to the material in terms of its color, acting as an adjective modifying the noun 'картон'. It specifies the type of cardboard needed, agreeing grammatically with it.
Для проекта мне понадобится немного коричневого картона.
For the project, I will need some brown cardboard.
In this sentence, 'коричневого' is used as part of the phrase 'коричневого цвета', which means 'of brown color'. It describes the attribute of the dresses ('Платья'), emphasizing their color.