Snowdin Forest is the stretch of subterranean trees that extend between The Ruins of Home to Snowdin Town. During the Underearth Arc, the area was inhabited with teen monsters.
BGM: Like Snowflakes
Snowdin Forest has almost completely been deserted. There are only a few monsters who live here now.
BGM: Snowdin Town
Snowdin Town is a town in The Underground that was founded in 1408 by a relatively large number of furry monsters. Since it's founding, it has acquired some interesting traditions (like putting gifts under a decorated tree). Sometime in the late 1900s, the Skeleton brothers, Sans and Papyrus, moved into a house on the far side of Snowdin Town.
Snowdin Town by this point is completely abandoned. Everyone who lived in Snowdin Town has left to live in New Snowdin.