This page is based on information presented in the cartoon, but is largely speculative in nature.
The core question that this page will attempt to address is very simple. What is "Magic" in Thundarr the Barbarian?
The answer is not simple. The short answer is: Magic is basically any source of energy that does not follow the known laws of physics.
A more precise definition is elusive due to the variety of sources of magical power seen in the show and the equally varied individuals who used them. Also confusing the issue is that there is no clear line between sources of power generated by advanced technology and magic, or even IF there IS a line seperating the two. While varied, the sources and wielders of magic can be placed into categories.
Sources of Power[]
Beings with innate power[]
The Abyss Wizard, Argoth, Ariel, Artemus, Circe, Dark Wizard, Khromm, Maya, Merlik, Mindok, Sarott, Sholow, Skullus, The Stalker from The Stars, and Striya each had powers that were apparently not derived from anything other than their own innate ability. There's a lot of differences in that group though. Striya and the Stalker aren't identified as Human at all. Maya's Element Shifting powers have only vague similarities to forms of magic that are normally seen in the show, but have some similarities regardless. For example, Maya's ability to turn to gas is similar to the method by which some wizards teleport. Is it the same?
Cyborgs[]
Chom, Gemini, and Octagon had visible mechanical components of their bodies. Chom never used his powers in his brief appearance. Gemini's nature was unclear. He was obviously a cyborg, but his power source was apparently internal. Octagon relied on an external power source to augment his abilities, but had some power even without it.
Beings who use an external power source[]
Many people in the show use magical artifacts, however most of them are not identified as Magic users, exceptions to this include Gorn, Infernus, Kublai, Valerie Storm, and Vashtarr. Each of them lost their powers when the source of their power was destroyed. The nature of the artifacts that granted their powers is unclear.
As mentioned above, there are many beings(Argoth, Circe, Dark Wizard, Khromm, Mindok, Octagon, Sarott, Striya) who use devices to augment their powers, even with innate powers. In some cases it's really not clear if magical weapons are sources of power themselves, or devices to channel a wizard's innate strength. For example: Argoth and Circe both used magical gems of some sort as spellcasting conduits. But in both cases the gem was seemingly a tool to focus magic, and not a source of magic. However Sarott and Octagon both had at one point devices that enhanced their abilities. Octagon's fortress had a energy reactor of some sort and Octagon could channel it's power to enhance his spells. Sarott wielded the Gauntlet of Power at times, and it did actually make him stronger. Allegedly it could make an ordinary man(who knew how to use it) capable of fighting a wizard. We don't get to see Gorn use it much though. He was seemingly no match for Sarott's powers.
Titles[]
It's also worth examining the various titles used for Magic wielders.
Witch[]
Witch seems to be a pejorative term in the series. No character in the series ever referred to themself as a Witch. When someone referred to Ariel as a Witch, it made her angry and she loudly protested that she wasn't one. Circe was often referred to as a Witch by others, but never used the word to describe herself. At one point Circe indirectly described herself as a Sorceress.
What distinguishes Witches from other magic users? The only character identified as a Witch is Circe. Circe's specialties were turning people to stone, turning people to frog monsters, and the body swap spell she used on Ariel. Few, if any, other magic users had these abilities.
Also, Witch does appear to be a feminine term. It was only ever used to describe Ariel (who found it offensive) and Circe. Mazem and Striya, despite being female, were not referred to with the word Witch, but with either Wizard or Wizardess.
Sorceror/Sorceress[]
This term was mainly used in its feminine form to describe Ariel. It was used directly to describe Merlik. It gets interesting when you look at indirect mentions. Argoth indirectly referred to himself as a Sorceror when explaining that his powers would (like Ariel's) weaken if he entered the jungle near his citadel. Circe specifically said that she needed the body of a Sorceress to swap with. Thus hinting that she considered herself to be a Sorceress. Then in Portal Into Time, Khromm is twice indirectly referred to as a Sorceror. On one of these occasions Khromm referred to his own magic as sorcery.
What distinguishes a Sorceror/Sorceress from other magic users? The only thing that Argoth, Ariel, Circe, Khromm, and Merlik all had in common was innate powers. None of them used magical artifacts as a power source. Argoth had a red gem that he used to talk to the Beast of the Forbidden Jungle, but this gem was apparently a tool only. Circe had a green orb she used in some of her spells, but it seemed to have no purpose beyond a magical conduit. But not all beings with innate magical power are referred to as Sorcerors/Sorceresses. Most are simply called Wizards.
There is a second trait identified as an aspect of sorcery. In many episodes Ariel is immobilized and thus can't use sorcery. When fighting Khromm, Ariel specifically stated that the key is to hold Khromm's hands. And when Thundarr pinned Khromm's hands it did in fact prevent him from spellcasting. Something similar happened when Ariel fought the Abyss Wizard. The fight was effectively over as soon as Ariel animated his throne to immobilize him. But, when fighting The Stalker from The Stars Ariel demonstrated that she had at least a limited ability to use magic with her hands webbed. But when captured by Argoth he was able to fully prevent her from spell casting with manacles attached to her wrists. Argoth also had an energy web that could do the same thing, so it could be the manacles were special somehow.
Wizard/Wizardess[]
This term seems to apply to almost anyone experienced with wielding magic. The Abyss Wizard, Argoth, Artemus, Chom, Dark Wizard, Gemini/Janus, Infernus, Khromm, Kublai, Mazem, Mindok, Octagon, Sarott, Skorpos, Sholow, Skullus, Striya, and Vashtarr were all referred to as Wizards at least once. Yando is worth an honorable mention as he tried to become a Wizard by collecting powerful artifacts. Wizardess was only used once in reference to Striya.
What did they have in common? Not much, some were normal or normal-ish looking people who simply had magical powers. Others were freakish looking and almost monstrous. Mindok's body was nothing more than a living brain. Some were powered by magical artifacts, others had enormous reserves of innate power. The only evident commonality is Magic power. Argoth, Gemini, Mindok, and Vashtarr were among the most powerful Magic users ever seen in the show. Ariel was not powerful enough to defeat any of them in single combat. Chom, Dark Wizard, Mazem, and Skorpos were not seen using their powers in combat. The rest were formidable opponents. Although Abyss Wizard, Skullus, and Striya were beaten in duels, they were not beaten easily.
Types of Power[]
This is something that gets mentioned in the show, but never explained in depth. Thus part of this is guesswork based on the bits we do know. However, there are a few eps that do at least discuss it to some degree.
Sorcery[]
One interesting thing to note in Island of the Body Snatchers is that Ariel learned some of Circe's abilities. Ariel uses sorcery, and... Circe seemingly does too. Circe targeted Ariel for the body swap spell due to the nature of Ariel's powers being compatible with her own. Then after Circe was petrified, Ariel used a magic ability she learned from Circe.
Enchanted Items[]
There are many "magical" relics in Thundarr the Barbarian. As mentioned above, Gorn, Infernus, Kublai, Valerie Storm, and Vashtarr seemingly had no powers without the objects that they wielded power from. Well... "magic" since we really don't know where they came from.
One thing notable is that Yando created the Nega-Sword, a powerful weapon, using... seemingly crude techniques. Would it be treated as magic by people who didn't know how it worked?
The helmet that's fought over in Challenge of the Wizards, well it's a metal crown found in the remains of a hydroelectric dam. It looks more like tech than the traditional magic crown, but.. we really don't know how it works. also it seemingly did little beyond acting as a power supply for spell casting. although is somehow gave the spells of the wizard using it's power an elemental weakness(can't affect silver).
The gauntlet used by Gorn, kinda looked like a piece of a suit of power armor. It has wires and circuitry everywhere. Also, it didn't seem to have actual powers of its own.