Besides the protagonist, the fallen human and Flowey, most characters encountered in the game are monsters. The game mentions a few real-world animals, such as mice, spiders, and dogs. It is unknown whether those count as monsters.
King Asgore Dreemurr rules all monsters in the Underground. After The War of Humans and Monsters, the humans from the Surface trapped the monsters underground. Despite the negative implication of the word "monsters," monsters in Undertale are not an evil species. It is even said that monster SOULs are made of love, hope, and compassion.
Monsters are by far weaker than humans, both in physical combat and regarding the strength of their SOUL. This inadequacy is because monsters are mostly made of magic, which in turn makes them attuned to their SOUL. The damage a monster receives from an attacker is dependent on both the monster's will to fight and the attacker's will to hurt. Monsters, however, can obtain an "unfathomable" amount of power when they absorb the SOUL of a human.
Monsters come in numerous shapes and sizes, although the notable species of monsters include skeletons like Sans and Papyrus, ghosts like Napstablook and their relatives and Boss Monsters like Toriel and Asgore. Monsters SOULs are upside down and wholly white.
A monster's body is mostly made of magic and is attuned to its SOUL. When a monster gets old, they enter a state called "fallen down." A monster in this state lies immobile and dies shortly afterward. When a monster dies, its body turns into dust. This dust is often spread onto the owner's favorite thing in funerals so that their essence lives on in their possessions. Monsters' bodies and SOULs are closely linked; when a monster dies, their SOUL is lost forever. Boss Monsters are an exception, as their SOULs exist outside their bodies for a few moments before shattering.
Because of their magical nature, monsters' bodies cannot handle high concentrations of determination. When monsters are injected with it, their bodies liquefy and melt together. The Amalgamates are a clear example of this.
- It is presumed that at least 322 monsters inhabit the Underground – given the number of monster souls visibly emanating from Asriel Dreemurr as the barrier is broken, plus incorporeal ghosts, who do not participate in the event. At least 104 monsters must be killed as part of fulfilling the requirements of the Genocide Route.
- Boss Monsters are very likely the only kind of monster that can live eternally. Using Asgore as an example, he fought in the war between humans and monsters and still lives many years later, existing when the first human fell and when the protagonist fell as well. An explanation for this can be found in one of Gerson's dialogues during the epilogue. 
- Love, hope, compassion... This is what people say monster SOULs are made of. But the absolute nature of "SOUL" is unknown. After all, humans have proven their SOULs don't need these things to exist. - Book 6 (from the left) in Snowdin Library
- Humans are unbelievably strong. It would take the SOUL of nearly every monster... ... just to equal the power of a single human SOUL. - Ancient glyphs in Waterfall
- While monsters are mostly made of magic, human beings are mostly made of water. Humans, with their physical forms, are far stronger than us. - Book in Snowdin Library
- Because they are made of magic, monsters' bodies are attuned to their SOUL. If a monster doesn't want to fight, its defenses will weaken. And the crueller the intentions of our enemies, the more their attacks will hurt us. Therefore, if a being with a powerful SOUL struck with the desire to kill... Um, let's end the chapter here... - Book in Snowdin Library
- When a human dies, its soul remains stable outside the body. Meanwhile, a monster's soul disappears near-instantly upon death. This allows monsters to absorb the souls of humans... While it is extremely difficult for humans to absorb a monster's soul. This is why they feared us. Though monsters are weak, with enough human souls... They could easily destroy all of mankind. - Monster History part 7
- Unfortunately, monsters are not experienced with illness. However, when monsters are about to expire of age, they lie down, immobile. We call this state "Fallen Down." A person who has Fallen Down will soon perish. In a way, this confusing situation was all too familiar. - Monster History part 6
- ASGORE asked everyone outside the city for monsters that had "fallen down." Their bodies came in today. They're still comatose... And soon, they'll all turn into dust. But what happens if I inject "determination" into them? If their SOULS persist after they perish, then... Freedom might be closer than we all thought. - Entry Number 6, True Lab
- Monster funerals, technically speaking, are cool as heck. When monsters get old and kick the bucket, they turn into dust. At funerals, we take that dust and spread it on that person's favorite thing. Then their essence will live on in that thing... - School report in the Snowdin Library
- There is one exception to the rules above: A certain type of monster, the "boss" monster. Due to its life cycle, it possesses an incredibly strong soul for a monster. This soul can remain stable after death, if only for a few moments. - Monster History part 8
- You see, unlike humans', monsters' bodies don't have enough... Physical matter to take those concentrations of "determination." Their bodies started to melt, and lost what physicality they had. Pretty soon, all of the test subjects had melted together into... Those. - Alphys
- ASGORE. He’s a real interesting type of monster. The strongest type of all. We call ‘em Boss Monsters. When they have an offspring, the SOUL power of the parents flows into the child... Causing the child to grow as the parents age. But ASGORE doesn’t have a child. So he’s been stuck at the same age... And probably will be forever. Huh? What if the child is a human...? What? Nah. That wouldn’t work. So if ASGORE’s your father, he’ll definitely outlive you. - Gerson, during the Epilogue