Each Creature Type includes many Species.
To understand the D&D Creature Types helps clarify what a "Species" is and how various Species group together.
Many Creature Types are planar, such as Fiend, Fey, and so on.
The main traditions that inform the D&D 5e planes are the 1e Players Handbook with its cosmology that includes a Plane for each Alignment Tendency and the 4e Players Handbook that features the Fey and Shadow planes.
1e Planes
The 1e cosmology maps the Positivity and Negativity as aspects within the Ethereal Plane, and maps the Alignments as Outer Planes that frame the Astral Plane. Meanwhile the Ethereal Plane encompasses, directly borders, and interacts with the Astral Plane. The worldview is an ethical creation, where Astral thoughts create Material matter.
The Astral Plane and the Ethereal Plane are levels of being. The Astral Plane is the level where things exist as thoughts. The aster is the stuff that thoughts are made out of. The Astral Plane is a realm of thoughts, relating to the Platonic Ideals, and behaving in dreamlike ways. The Ethereal Plane is the level where thoughts begin to physicalize and influence. The ether of the Ethereal Plane is the fifth element, from which the four elements derive. The four elements are matter, having mass. Altho the ether is physical it lacks matter. From a mythic perspective, the ether is spirit especially as the substance that spirits are made out of. For example, the ki is a personal aura of spirit that extends out from the body to surround the person with an area of ethereal influence. From a scientific perspective, the ether is force, especially the fundamental forces that all things are made out of: gravitational force, electromagnetic force, and the nuclear forces. Even matter itself is ultimately made out of immaterial force. These elemental Ethereal forces permeate the Material Plane and emanate like an aura of influence. Within the ether, the four elements potentially exist.
The Astral Plane directly interfaces the Ethereal Plane. They flow into each other. Thoughts physicalize into reality. Reciprocally, reality inspires new thoughts. Telekinesis works this way: thoughts manifest force. The spiritual bodies of souls are conscious.
The Astral, Ethereal, and Material Planes overlap. The Ethereal overlaps the Material, and the Astral overlaps both. In the 1e map, the Astral Plane has the Ethereal Plane at its center. The Astral Plane also overlaps this Ethereal area too. Something can exist at all levels of being simultaneously. A Material body of matter, can have an Ethereal aura of lifeforce, and an Astral consciousness that visualizes a mental construct of ones self-image.
(In 5e, a "Wildspace" is a region of the Astral Plane that overlaps and simulates a solar system in the Material Plane. There is a virtual reality version of that solar system made out of Astral though. This Wildspace then overlays the Material Plane as an augmented reality, where objects made out of aster can coexist sidebyside with objects made out of matter.)
The Positive Plane and the Negative Plane are present within the ether. Positive influences move the four elements of Material matter to interact with each other constructively to bring forth and sustain life. Oppositely, Negative influences move these elements to interact destructively ending life.
In 1e, Astral thought, Ethereal spirit, and Material matter are an overlapping and interlinking great chain of being.
3e Planes
3e evolves from the 1e cosmology albeit convoluting with more details, sometimes conflictive, where related concepts disconnect from each other, and resulting in a sometimes bewildering sprawling big picture. Here is a popular map by Zen79 attempting to track of all of the 3e planar settings while striving to adhere to the official maps as much as possible except where official texts contradict the official maps. Here, what would officially be the "Prime" Material Plane appears instead as the planet Toril of the Forgotten Realms setting. Thus the assumed primacy decenters to accommodate whichever Material Plane setting a group chooses to use. A main difference between 1e and 3e is, in 1e the Elemental Planes are aspects of the ether that surrounds the Material Plane, but in 3e the Elemental Planes split away from the Material Planes, to form their own independent region. Meanwhile, the Positivity and Negativity function as if new elements. The relationship between the four elements of matter and the Material Planes of matter obfuscate. Meanwhile, the assignment of a True Neutral Alignment Plane locating at the center of the other Alignments displaces and divorces the Material Plane that was here in 1e. Where once was matter, is now the Astral City of Sigil.
4e Planes
4e consolidates the 1e and 3e cosmologies into a multiversal Axis. This Axis strings out the 3e continuum from the Elemental Planes to the Material Planes to the Astral Planes, with the Astral upright as the upper pole and the Elemental as the Lower pole. Now the Material Plane is not simply matter, but is in the middle as mix of Astral thought shaping Elemental matter. The Astral Plane is a celestial sea with the Alignment Planes forming its islandlike stars. Into this continuum, 4e reinterprets the earlier Positive Ethereal and Negative Ethereal as the Plane of Fey (aka Feywild) and the Plane of Shadow (aka Shadowfell). These are the ethereal "echoes" of the Material Plane.
The 4e design goals strive to consolidate similar gaming themes into one salient go-to place.
The Fey Plane of Feywild (originally an obscure plane called Faerie) and the Shadow Plane of Shadowfell (originally relating to Illusion) already existed in 1e but off-the-map. 4e reinvents them as central conceptual themes within the 4e Cosmology. Fey is the go-to for the themes of fairytales and folklore, and Shadow the legends about the underworld realm of the dead. Together the Fey and Shadow replace the earlier Ethereal Plane. To teleport or move incorporeally, one traverses immaterially via Fey or Shadow, rather than via the Ether.
The 4e Primal Power Source proved popular. It explores animistic shamanic themes, focusing resolutely on the life and landscape of the Material Plane, often called the Natural Plane or Nature. While Fey and Shadow also participated in these Primal themes, the status of these two Planes remaines ambiguous: whether they are Primal thus Natural, or Arcane thus Unnatural.
The some 4e consolidations fare less popular. By awkward conflation with Shadowfell, 4e Illusion magic is deathly, gloomy, and ghostly.
Highly unpopular, the 4e Alignment system consolidated the Alignments down to five: namely LG, G, U (Unaligned), E, and CE. It omits the concepts of Chaotic Good, and makes "obedient" Lawful Good the "best" kind of Good. The possibility of "rebellious" Good or a "protest" for the sake of Good become erased. Meanwhile the absence of Lawful Evil make the dysfunctional and abusive uses of the Law uncritiquable. The 4e "Chaotic" fight for individualism and freedom was declared as if always Evil, and moreover the "worst" kind of "Evil". Meanwhile, this conflated continuum stretched across the cosmic Axis, with Lawful Good at the top, Unaligned Sigil in the middle neigboring the Material Plane, and exiling the Chaotic Evil Alignment Plane to the bottom, far away from the Astral realm of thought into the Elemental matter. By implication, matter itself becomes as if inherently Evil. 4e did much objectionable with regard to the D&D Alignment traditions. But it also made Alignment strictly narrative without any mechanical implications. Thus in practice, D&D gamers easily ignored the 4e Alignment problematics.
The 4e Axis cosmology omits the Positive Plane and the Negative Plane from 4e. These instead consolidate as damage types, Radiant and Necrotic, that roil as part of the Elemental Chaos. Yet, the existence of an Axis in itself implies a transcendent Positivity orienting the Astral pole upward, and a transcendent Negativity orienting the Elemental pole downward. The entire cosmology polarizes between Positive as Goodness and Negative as Evilness.
5e Cosmology
The 5e cosmology appreciates and preserves much of the 4e cosmology, while abandoning some difficulties, and reorganizes the Axis under the influence of the original 1e cosmology for a more concentric framework. The Material Plane is the center of a mandala-like circular multiverse.
An understated yet profound innovation of the 5e cosmology is the location of the Energy Planes, Positive and Negative. These are transcendent principles that orient the entire multiverse.
• The Positive Astral is the Celestial Planes (aka Upper).
• The Positive Ethereal is the Fey Plane (aka Feywild).
• The Negative Ethereal is the Shadow Plane (aka Shadowfell).
• The Negative Astral is the Fiend Planes (aka Lower).
Partly the reogranization of the Energy Planes derives from thinking more carefully about cosmological theory. Partly it is practical. Under the influence of the Cleric class, which coheres with an Astral Plane and with Positive Healing, there seems some kind of connection. Meanwhile, Celestial creatures such as Angels seem inherently Positive. Hence the Positivity permeates all that exists, including the Astral Plane.
Strictly speaking, there is only Positive Energy. The Negativity is an absence of energy. It restricts (hides, dampens, unravels, or voids) the Positive Energy. The term "Negative Energy" is an improper turn of phrase for empty Void.
At the Astral level of existence, the energetic influences are "Positive thoughts" and "Negative thoughts". This can and does relate to the D&D ethical Alignment system. With important exceptions, Positive thinking creatures are likely to exhibit compassion and do Good, and Negative thinking creatures likely to exhibit judgmentalism and do Evil.
At the Ethereal level of existence, the energetic influences are active forces and influences within the spirit realms to promote life and abundance. Here the energetic influences have less to do with Alignments. Evil Fey and Good Undead are common enough. The Positive ether supports and encourages life. The Negative ether undermines and extinguishes life.
At the Material level of existence, the energetic influences are more Neutral. Positive packets of energy whirling within the Negative vacuum of empty space, making creation possible. The resulting matter can be tools for either Positive or Negative purposes.
The Elemental Planes
The 5e Elemental Planes locate awkwardly in the 5e cosmology in the context of the D&D traditions. 3e and 4e cosmologies had split the elements away from the Material Plane. 5e now wraps them around the Material Plane like 1e, but now the Elemental Chaos separates the Material Plane from the Astral Plane. The resulting situation appears as follows:
• There is no longer a direct interactive border between the Astral Plane and the Ethereal Plane.
• The Elemental Planes are the border between Astral and Ethereal.
• The Ether separates the elements away from the Material Plane.
• The Elemental Planes feel immaterial, far from matter, and more diaphanous and dreamlike between Ethereal and Astral.
Are Elemental creatures a "form" without "substance"? Namely spirits but not matter?
5e Creature Types
What is an Elemental species? It depends on the planar cosmology. Understanding what the Elemental Planes are, exactly, directly impacts what other Creature Types are as well: Celestial, Fiend, Fey, and others, because of how they do or dont relate to the Material Plane.