Koopas are a race of turtle-like creatures. There are many different types of Koopas, the best-known of which are the Koopa Troopas.
Most Koopas seem to operate in a hierarchy-based grouping system. Bowser is the leader of the militaristic Koopa Troop, an organization that has attempted to conquer the peaceful Mushroom Kingdom many times and which has control over most of the Koopas in the Mario series. Many of these same Koopas are soldiers in this army, and have different tasks in the organization based on the abilities of their particular class. For example, Koopa Troopas tend to be assigned as basic foot soldiers alongside Goombas due to their lack of special abilities, while Magikoopas are often given advisory roles due to their magical abilities and gift of foresight.
However, not all Koopas are allied with Bowser. Some Koopas, such as the Koopa Troopas of Koopa Village, live peacefully inside the borders of the Mushroom Kingdom, while others, such as Koopa the Quick in Super Mario 64, simply hold no allegiance and operate independently.
Similar to turtles, Koopas possess shells that are usually a defense mechanism for the reptiles. Unlike turtles, however, the shells are not integrated into their body and more-or-less act as wearable armor/clothing, as evidenced by Super Mario World and Paper Mario.
Koopas are probably the most diverse class of enemies in the Mario series, with dozens of Koopa variants having appeared since their Super Mario Bros. debut.
Beach Koopas are shell-less Koopas. They either wear a shirt or boxer shorts. Usually, Beach Koopas search for their shell, even if Mario destroyed it.
Bombshell Koopas are Koopas who live in Sarasaland in Super Mario Land. Unlike most Koopas, this variant seems to be under Tatanga's command; none are known to be affiliated with the Koopa Troop. When attacked, Bombshell Koopas vanish and leave their Shell behind to explode.
Boom Booms are a Koopa species similar to Bowser's. Booms Booms are very large in size and most have spikes on the back of their shells. Their most notable traits are their huge arms, which they use to attack by flailing them around. Members of this species include Boom Boom and a female Boom Boom named Pom Pom.
Not much is known about the species of Koopa to which Bowser belongs. The only members appear to be Bowser, Bowser Jr., the Koopalings, and the Koopa Kids. Their most notable feature is their spiked shells, similar to a Spiny. Unlike a Spiny however, members of this species of Koopa stand upright, and are much larger than most types of Koopas upon reaching adulthood. Even as a child, Bowser Jr. is almost as tall as a full-grown Koopa Troopa. Another unique feature of this species is that their shells don't entirely cover their whole body. Instead, their shells only cover their backs while only having a padded stomach in the front(that could be part of their shells). These Koopas are known for their strength, sometimes the ability to breathe out flames, and durability, even as children, as well as magical powers; they typically are the leaders of the Koopa race.
Chargin' Chucks (sometimes referred to as Koopa Quarterbacks or Koopa Football Players) are a type of Koopa who wear American football gear. They have a variety of attacks and sports based weaponry, such as damaging baseballs and footballs. Chargin' Chucks first appeared in Super Mario World. Like the Hammer Bros. and their variations, Chargin' Chucks often appear as duos, or sometimes even larger groups.
Dark Koopas are Koopa Troopas with purple shells that only live in underground places, for example, the Toad Town Tunnels. They are only seen in the games Paper Mario and Paper Mario: The Thousand-Year Door.
Dry Bones (also known as Skeleton Koopas) are skeletal versions of Koopa Troopas introduced in Super Mario Bros. 3. These undead Koopas will often collapse when attacked, but they can reassemble themselves after a short while and continue to battle their opponents. A rarer, larger version exists: the Super Dry Bones.
Electro-Koopas are Koopas found in Super Mario Sunshine. There are three kinds of Electro-Koopas, all of which walk on all fours and have electrical shells which they can throw like boomerangs. From what has been seen in the series so far, Electro-Koopas are native to Isle Delfino.
Gargantua Koopa Troopa
Gargantua Koopa Troopas are a type of Koopa Troopa native to Big Island and the level Jungle of the Giants, as depicted in Super Mario Bros. 3 and New Super Mario Bros. U, respectively. Like many creatures found on Big Island, Gargantua Koopa Troopas are enormous versions of their counterparts in the rest of the world. These gigantic Koopas are also found in Super Mario Galaxy 2's Supermassive Galaxy.
Hammer Bros. (or Hammer Brothers) are a type of Koopa that stand upright, wear helmets and throw hammers at passerby. There are various similar Koopas who throw other objects at their opponents, such as the Boomerang Bros. and the Fire Bros., the latter of which spits its projectiles from its mouth, instead of pulling them from thin air. All of these Koopas have similar shell patterns, and beaks that are lighter in color than the rest of their face, both of which sets them apart from Koopa Troopas, which are also slightly smaller in size.
Koopa Paratroopas (also known as Parakoopas, Sky Troopas in Super Mario RPG: Legend of the Seven Stars, or simply Paratroopas, in Japanese パタパタ Patapata), are Koopa Troopas with wings. They can bounce along the ground or hover back and forth with their wings, which will fall off if the Koopa Paratroopa takes damage, turning it into a normal Koopa Troopa. It is unknown if de-winged Koopas can re-grow their wings, though it appears that not all Koopa Troopas can grow wings in the first place, suggesting the Paratroopas are a distinct species. They were first introduced in the original Super Mario Bros. and have been a staple of the Mario series ever since.
Koopa Kids are a species of small Koopas that look similar to Bowser and work for him. They have appeared in every Mario Party game except Mario Party DS, Mario Party 8, Mario Party 9, and Mario Party: Island Tour. In those games, they were replaced by Bowser Jr.. They come in 3 colors: Red, Blue, and Green (green is the one that is most commonly seen). In spite of what their name implies, these Koopas do not appear to be Bowser's offspring, and their exact relationship to Bowser has never been disclosed.
Koopa Troopas (often shortened to plain Koopas and occasionally to Troopas, Japanese ノコノコ Nokonoko) are common enemies in the Mario series. While some are peaceful, Koopa Troopas are usually displayed as foot soldiers of the Koopa Troop under the direct leadership of Bowser. They are considered to be the archetypal Koopas, as they have no special abilities like Hammer Bros. or Lakitu. Throughout the Mario series, Koopa Troopas have appeared as both quadrapedal creatures (four legged) and bipedal (walking on two legs). Since their debut in Super Mario Bros., Koopa Troopas have appeared in almost every game, usually as enemies, but also as playable characters or party members.
Koopatrols are armored Koopas with spikes and appear to be Bowser's main guards. They first appear in Paper Mario.
Koopeleons are a rare and unique sub-species of Koopas that live in the Toadwood Forest in Mario & Luigi: Partners in Time. A Koopeleon resembles a Koopa and chameleon hybrid with a blue shell, green skin, large eyes, and a long curled tail. Koopeleons have the ability to turn invisible, a survival mechanism that allows them to hunt unsuspecting prey. Gold Koopaleons are practically the same thing, but live inside Thwomp Volcano and have an extremely high defense.
Lakitus are bespectacled Koopas who ride clouds through the skies, and which can occasionally be found snorkeling in water and hiding behind walls or in pipes. Unlike most Koopa species, they do not have snouts or beaks, but flat faces; their shells are also unique in their circular patterning. To attack, Lakitus typically throw unlimited numbers of Spiny Eggs at their opponents, and some will also dangle 1-Up Mushrooms from fishing poles to lure their targets closer.
Magikoopas (referred to as Koopa Wizards in the Super Mario World television series) are a variant of Koopa that can cast spells and fly on broomsticks. They are generally considered to be high-ranking members of the Koopa Troop. They wear glasses and wizard's garb; typically blue robes and hats, though Magikoopas specializing in different forms of magic sport different colors. Aside from their heads, which have slightly larger beaks than those of their normal counterparts, little is known about the Magikoopa's external structure due to their robes.
Robot Monster Troopa
Rocky Wrenches are enemies that resemble Monty Moles and pop out of manhole covers on airships at random. They were considered a type of Koopa at the time of Super Mario Bros. 3,  but they lost their shell after their debut appearance and may have been reclassified as a consequence.
Snooze-A-Koopas (also known as Snooza Koopas) are bizarre-looking Koopas encountered by Mario in the game Super Mario Sunshine. Snooze-A-Koopas have flippers instead of hands or feet and walk on all fours; they are also quite large and have a shell with a Yoshi Egg-like pattern on it. They spend much of their time sleeping, but they can be violent when provoked.
Spikes are small Koopas who either stand still or walk around. When Mario or an ally of his comes by, however, Spikes throw spiked balls to harm the character. More specialized variants, such as Snow Spikes and Stone Spikes, also exist and throw their namesake projectiles instead of spiked balls.
Spinies (Japanese トゲゾー Togezō) are four-legged Koopas known for their spiked shells. They appear to have a symbiotic relationship with Lakitus, another Koopa variant which throw Spiny Eggs at their opponents.
Super Troopas are large and strong Koopa Troopas that are very dull and clumsy. They serve as Bowser's guards and only appear in the The Super Mario Bros. Super Show.
Super Koopas are shell-less Koopas that wear a cape, allowing them to fly.
Terra Cottas are advanced versions of Terrapins they are only found at Bowser's Keep in the game Super Mario RPG: Legend of the Seven Stars. Instead of having green shells like Terrapins, they have red ones.
Terrapins are armored Koopas with green shells that appear in Super Mario RPG: Legend of the Seven Stars. In spite of their armored appearance, these Koopas are the weakest enemies in the game, even weaker than Goombas.
Names in other languages