This is the 12382nd most frequent Russian word.
каретой
"Каретой" means "by carriage" in English.
Here, "каретой" is used in the instrumental case to describe the means of transportation, i.e., the carriage.
Мы поехали в город каретой.
We went to the city by carriage.
In this sentence, "кареты" is the genitive singular form, specifying "of the carriage" as the source or origin.
Она вышла из кареты и посмотрела вокруг.
She stepped out of the carriage and looked around.
Here, "карета" is in the nominative case and is the subject of the sentence.