It's pretty consistent
Creatures can be
- creatures in the common sense of the word, often harmless and small
- monsters (which have stat blocks). Humanoids are a subtype of monster
- player characters (which have race, class and levels on character sheets)
Characters can be
- player characters, only controlled by the players
- nonplayer characters, commonly controlled by the DM
Characters are about the role someone plays in the story and about having a personality. Not all characters must be creatures, although nearly all are, and the vast majority are humanoids, too. Not all creatures are characters, most monsters and harmless creatures are not.
The core player character races are humanoid, too, but PCs are not generally considered to be monsters, which are the DMs domain.
The game uses the plain English reading for game terms when no specific mechanical definition is supplied, as is the case here. The dictionary primarily defines creature as
any large or small living thing that can move independently
There are a large number of creatures that are too harmless to be categorized as a monster (even of challenge rating 0), and for which the game does not provide game statistics, such as individual small insects, earthworms, puppy dogs and baby kittens.
These are not inanimate and therefore not objects under the game's rules, nor are they points in space. If they were not creatures in game terms, you would not be able to attack them, target them with spells like invisbility, or damage them with area effects like fireball.
Everyday things in D&D 5e work the way we expect them to, and such mundane, everyday creatures therefore need to be creatures under the game's mechanics.
However, this can also causes issues: some spells affect a limited number of creatures or are triggered by creatures. Do you want your Alarm to go off because of an ant? Tiny Hut to fail due to meal worms in your flour rations? Rope Trick's space blocking you due to lice in your hair? A canary you carry in a small cage to detect poison gas or live crickets in a jar to feed your familiar being left behind when you misty step? A good rule of thumb for play is to just ignore harmless, diminuitive creatures unless you care about them.
The game is fantastic, and it is not limited to everyday living creatures. Both monsters and player characters (which have many unusual abilities) are creatures, too.
For monsters, the Monster Manual says on page 4:
A monster is defined as any creature that can be interacted with and potentially fought and killed.
So clearly, monsters are creatures. Monsters are grouped into different types (MM, page 6), like beast, giant, or humanoid. Some of them are not living, but still are creatures, for example those of undead or construct type. Some of them may have a subtype tag (MM, p. 7), for example humanoid is broken down into human, elf, orc, goblin etc.
The rules don't actually state explicitly that player characters are creatures, as this is all too obvious. It's so obvious that it feels a bit silly to demonstrate (which is likely why the rules omit it):
- The rules for creatures in the combat chapter of the PHB only make sense if the player characters are creatures. For a concrete example, the language of the Help action states "You can lend your aid to another creature", "another" implying that "you", the player character, are a creature too.
Nonplayer Characters (NPCs)
The DMG defines NPCs on p.89 as
any character controlled by the Dungeon Master
In this context read character as "anyone that plays a part in the game's story". Most NPCs are creatures, but NPCs are not limited to being creatures. Sentient magic items function as NPCs (DMG p. 213) and are objects, not creatures. There are also a lot of creatures that are not NPCs, including the PCs and most monsters. Appendix B of the Monster Manual lists humanoid monsters that are NPCs. Im sum, NPC is an independent concept that is overlapping with creature, but cares more about personality and story.
Both Player Characters and Nonplayer Characters are characters.
In the PHB, the term is synonymous with Player Character, as the Index on p. 315 says:
player character. See character
In the DMG it often refers to player characters, sometimes as above to characters in general (both PC and NPC), or just to NPCs, so the use of the term is not consistent. Most of the time however, you can tell from the context what is meant.
The determining characteristic of creatures that are dangerous enough to be fought is that they have full game statistics, including size, ability scores, speed, hit points, armor class. For monsters, these come in form of a stat block and include a challenge rating, for player characters, they come in the form of their character sheet, for creature NPCs it can be either.
There are a number of entities in the game that share something with creatures and may have partial game statistics beyond objects hp and AC, but are not creatures under the game rules, which often causes questions. For example (the list is not exhaustive):
Souls or spirits. In rare cases souls and spirits can become creatures, for example the Ghost is an undead monster that also is a soul. In nearly all cases, however, creatures have souls (DMG p. 24), so the creature consists of the soul together with the body. The soul by itself is not a creature, and what it exactly is remains undefined.
Unseen Servant. This spell creates a shapeless force that shares some attributes with a creature, such as having AC, 1 hit point, speed and Strength, but it is not a creature, lacking other attributes and a full stat block.
Eldritch Cannon. This feature of the Artificer Class (from Tasha's Cauldron of Everything) shares some attributes with a creature like having an AC, hit points, size, and speed, but it is an object, not a creature.
The shapes created by Shape Water. They can have the form of a creature, but lack any of the other statistics, being merely shaped volumes of water.