Creature is the general term to describe sentient beings in Dungeons & Dragons 5th edition. There are 15 types of creatures in this edition, differing slightly from previous editions. It's important to note what type of creature an opponent is because many spells and abilities only work on select types.
The origin of an aberration's magic is often mysterious, although sometimes they occur because of some kind of magical corruption. In the end, they tend to be alien to the way we understand the magic of our world. Many act on the bidding of a master, like the spectator, and others are independent beings, like the mind flayer, that rightly strike fear in the simple adventurer. They're especially common in official campaigns and high magic settings.
Beasts are usually animals you can find in the real world, but also includes dinosaurs. A few have magical abilities, but most are on the unintelligent side and are commonly used as pets, familiars, and mounts. On the other hand, giant beasts, like giant spiders, are classic enemies.
Near god-like, celestials are heavenly or legendary creatures like angels and unicorns which come from the upper planes of existence. They are the thematic opposite of Fiends and thus are usually good aligned.
A construct is made when magic is used to animate something not otherwise alive. Animate mud and you get a mud golem. Armor and swords can also be animated, though most things tend to take a more humanoid form. While most are created here on the material plane, some do come from other planes of existence. Constructs typically have no mind of their own, following specific orders given to them by the spellcaster that made them until they complete the task or are done in by time.
The game's namesake, the dragon, is a tremendous lizard with wings. True dragons fall into one of two categories: metallic and chromatic. This describes their color as well as their typically polar alignments of good and evil, respectively. A dragon's power increases as it ages, so it's important to note if the one you come across is a wyrmling, or if it's young, adult, or ancient. Relatives of the dragon, like wyverns and faerie dragons are also under this type.
An elemental is similar to a construct in that its life force is magical, however, an elemental exists because of natural magic inherent in the universe and tend to come from elemental planes of existence. They are the personification of a specific facet of nature, whether that's wind, fire, or something else. Some are more fiendish, like mephits, some resemble animals, like the salamander, and some others are quite human, like the efreeti.
Fey are another type of magical creature. They're strongly tied to the natural world and thus are often chaotic aligned. More often than not a fey is legendary or Tolkien-esque in origin, having ties into ancient Pagan lore of the real world, such as satyr and pixies. On the other hand, there's the blink dog, which is pure Dungeons & Dragons.
Demonic, devilish, and nightmarish, fiends come from the hellish, or lower, planes of existence. Unless you find yourself on one of their planes, these beings are usually summoned by a powerful spellcaster. They have strong urges of their own to corrupt and destroy; few are powerful enough to summon and control them. Occasionally, fiends are even sent to the material plane by the gods themselves. Common ones are imps, succubi, and hell hounds.
Like humanoids, but taller, giants tower over their bipedal brethren. Some are nearly as tall as the clouds they're named for, but there are also smaller varieties such as the ogre and the ettin.
Put simply, humanoids are any type of creature that is bipedal and resemble the human species in that they have a torso and four limbs, though some have stronger ties to other creature types and are therefore categorized as such. They tend to be civilized and relatively intelligent and thus are some of the most common non-playable characters and enemies, and almost all the playable species.
Monstrosity is a catch-all term for what doesn't fit into other creature types. Many of them are chimeras like the griffon and legendary monsters like medusa. This is also where the shapechangers fall, like mimics and doppelgangers. Their origins are equally varied, many being the result of magical experimentation and others became what they are because of terrible curses.
An invertebrate creature, the ooze is amorphous with absolutely no rigid structure in its body, unless, of course, it has recently eaten. They tend to live in dark places like caves and deep dungeons, nearby more violent creatures in the hope of feeding on whatever they leave behind, whether that's discarded animal bone or an injured adventurer. The gelatinous cube is an iconic ooze, and the black pudding is also a classic.
These aren't ordinary plants, but creatures that move and are often carnivorous. They are often drawn to the corruption of nature and the blood of adventurers. Some plant-type creatures are even good and highly intelligent, like treants.
Undead creatures used to be living, but through magical means have become corporeally or spiritually reanimated. They don't usually retain the mind and will they had in life, but most often are fixated a single personal task or the will of the one that reanimated them. Common undead are vampires, ghosts, and zombies.