The Underground

The Underground is where Chapters 1- 3/4 of Chapter 6 take place. It contains 5 main areas and 2 sub-areas. It's located underneath MT. Ebbot.

The Ruins
The Ruins is the place where Monsters, when they were first sealed Underground, lived. However, after 'Home' got over-populated due to its small size, monsters began expanding outside of the Ruins, leading to only a small amount of Monsters to live in the Ruins. Examples of said inhabitants are Froggits, Whisumns, and Loox. It's sometimes referred to as a catacomb.

Snowdin/Snowdin Forest
Snowdin is an area of the Underground where it is extremely cold close to all the time. It is accessible either by going through the Ruins or Waterfall. It is mostly inhabited by dog-like Monsters, well, really any monster with fur.

Waterfall
Waterfall is a place with an average temperature of around 60 or 70 degrees Fahrenheit. It is the perfect place for fish-like monsters or simply monsters who don't like extreme heat or cold to live in. It is also where Monsters find a lot of items from the surface that fell down there.

Hotland
Hotland is extremely hot due to being close to the Earth's Core, which is also why magma is close to everywhere. It is a place mostly inhabited by monsters who can withstand the heat.

The Core
The Core is a huge powerhouse that gives the Underground it's electricity. Nobody lives here due to magical electric hazards.

New Home
New Home is the capital of the Underground. It, like Waterfall, is a place that has a stagnant, hospitable temperature. Half of the population of the Underground lives here, including King Asgore.

Trivia

 * Rose lives in the Ruins with Toriel.
 * Snowdin is cold because it is in a part of the Earth that is not nearby a powerful heat source (e.g. the sun, the Earth's core, huge amounts of lava in a single area)