"Natsukashii" is a Japanese adjective referring to feelings of nostalgia for the past, sometimes with connotations of nostalgia for an idealized past that never was.
It is a word that is difficult to translate into English, which may account for why it appears so frequently around the internet in its original Japanese. This difficulty arises because while "natsukashii" refers to nostalgia it is not used in the same way that the English word "nostalgic" is used - whereas "nostalgic" refers to the person who is remembering, "natsukashii" refers to the thing they remember. The closest translation I can think of would be the English phrases "dear old" and "good old", as in:
dear old home
old favorite song
dear old friend