This is the 7557th most frequent Russian word.
дорогие
"Dear" or "expensive," depending on context.
In this sentence, 'дорогие' conveys an emotional closeness or affection, used as a form of endearment towards friends.
Дорогие друзья, рад вас видеть.
Dear friends, I'm glad to see you.
In this context, 'дорогие' (shortened to 'дороги' for the plural form of the adjective to agree with 'воспоминания') means 'precious' or 'valuable' in a sentimental sense, not monetary.
Эти воспоминания мне очень дороги.
These memories are very dear to me.
Here, 'дорогие' refers to high monetary value, describing jewelry that is costly.