A thaum is an elementary measure of magical energy, like a joule or a watt or a volt. The Mage Council defines 1 Thaum as the magical energy to heat 1 cubic half-meter of water 1 degree Celsius - not that they use those terms, mind you, but there you go.

Thaum is not discreet, however - it is a field, and one that is generally difficult to divide or otherwise separate. Thaum has a natural tendency to flow through and permeate everything. Individual objects and creatures have their own thaumic contents which is distinct and unique to that object - as thaum enters it, it takes on the thaumic pattern of the object, and when thaum leaves, it returns to it's previous state. This field isn't limited to a planet, but exists throughout all known space and time. In addition, most living things naturally produce thaum. This does, of course, beg the question of "will the universe 'overflow' somehow?" It is a question many academics have tried to answer, but no cogent response has been reached so far. So far, the question has been ruled interesting, but not of actual importance - the thaumic pressure hasn't noticeably changed in the thousand years of measurement.

There are 2 ways to think about thaum - the way it is perceived, and how it behaves. First is the matter of perception - and in this sense, thaum is thaum. The thaum of water is the same thaum in fire, and making one into the other is no more difficult than breathing. Forcefully, and there is a pretty pattern involved, but the point remains. The thaum on the planet is the same as what is in the outer spaces beyond, and you can watch the transition from one to the other, if you get close enough.

The second is behavior - thaum seems to naturally exist in certain states independent of pattern. Most thaum you naturally engage with is "terrestrial" - thaum of the natural world. Living beings produce it, for example, including the planet itself. It maintains certainly rules - like gravity, the relationship between velocity, acceleration, and time, and even time itself - perhaps with the assistance of the gods, or perhaps as a natural occurrence.
Celestial thaum, by comparison, is transformative, inconstant, and non-rules based. It can do things that make no sense when considered in terms of terrestrial magical laws. For example, rituals don't mean anything in terms of terrestrial magic - rather, it is the mindset that a ritual invokes that assists in casting. Swearing an oath to only use binding magic when wearing chains means nothing, as another example. A mage may be better at casting binding magic while wearing chains, but this is either the result of having a clearer image to keep in mind, or a built-up mental relationship, making the necessary pattern easier to recall due to the touchstone of the chains. Celestial magic, in comparison, is riddled with ceremony, oaths, and offerings - and these things matter a lot. They aren't exactly necessary, but the impact of an ordering mechanism seems to enhance magic using celestial thaum immensely.
It is important to know that Celestial thaum is terribly powerful, and inherently corruptive to spell patterns. While using Celestial thaum may seem like an easy and powerful way to accomplish...just about anything, honestly, it is important to remember that using it requires significant concentration and power to stabilize a pattern when this form of thaum is applied. A single second of lost thought would turn a normally safe transformation into an abomination ending in a merciful death. With few exceptions, it is far safer to wield a different form of thaum. This of course assumes that the practitioner has an option, of course - notably, clairevoyance, time sight, and prophecy can only be managed through Celestial thaumaturgy - dreamwalking and angelic summoning are similarly made possible through Celestial magic (the latter due to the conflicting requirements of needing to open a portal, a traditionally necrotic spell, into a fundamentally vivetic realm, without detonating in the conflicting energies - Celestial thaum's transformative nature allows for a similar portal to form, but with a less combustible mix)

There are 4 additional "forms" of thaum known - necrotic and vivetic, spiritual, and neutral.

Vivetic thaum is sometimes called "order" thaum, and necrotic is called "chaos" thaum - this is not correct, but it isn't a terrible way to think of it. Vivetic thaum maintains things. A wall infused with vivetic thaum, even unpatterned thaum, will survive longer, maintain it's shape better, and generally just exist better and longer. Necrotic thaum is similarly thaum that breaks things - but this can include things like "laws" that terrestrial thaum usually enforces. Most living beings contain both vivetic and necrotic thaum, and suffer if either is forcibly suppressed, though vivetic is far more abundant.

Spiritual thaum is weird. It seems to only occur in sapient beings, and seems to have very few natural uses except for being a sapient being. The gods are by far the largest concentration of spiritual thaum that exist, collectively. Ghosts are also almost pure spiritual thaum. "Inflicting" spiritual thaum on non-sapient beings has had mixed results...VERY mixed results. Golem intelligence has been produced via very careful applications of spiritual thaum, for example - but so have sentient plagues, land masses that have "decided" to move, killing thousands or even millions (the archeologists are working on it), and any number of equally bizarre occurrences. And, even if you do create some form of sapience, nothing remotely guarantees it will be a sapience similar to a beings, or even a particular creature. Nor is there any guarantee that it will be a stable awareness. It is also known that trying to press raw spiritual thaum onto a demon or angel is axiomatically lethal to those beings. Using a half-demon or half-angel as a pattern has better results...in that most such affected beings don't poof out of existence instantly. However, every single one of these "successes" have used their newly found self-awareness to seek out the fastest means of suicide available.

Neutral thaum is a bit of a misnomer. It is, rather than a "form" of thaum, an absence of form. It is entirely inert, incredibly hard to produce, and scares the living minds out of every god that's seen it. It never occurs naturally, as far as all research shows.* In addition, unless carefully contained, neutral thaum will take on a thaumic form given any sort of "push" - most often spontaneously turning into celestial thaum. It is theorized that neutral thaum could be used in a spell and actually have an effect on dragons...if you could a) get the thaum into the thaumic pattern, and b) get the resulting spell to do anything. Most researchers treat this endeavor as a useless wild goose chase.

There are other, theorised thaumic forms - the most hunted for is 1 or 2 forms related to thermodynamics and matter states. These theoretical forms will be covered in depth in course 453: Theortical Thaumic Formations.

Now, these two perspectives seem very very different, but are fundamentally both correct. As implied in our discussion of neutral thaum, a particular volume of thaum may be manipulated into taking a different form, and terrestrial thaum, for example, not only naturally converts into vivetic and necrotic thaum, but also can maintain a stabilizing effect that keeps the normally explosive interaction between the two into something as common and unconsidered as your digestive processes, which breaks down matter in order to create sustaining chemicals and energy. Thaum is, indeed, thaum, even though it can act in decidedly different ways, and it's interactions can vary wildly.

For additional reading, please consider the following: The Holy Treatises of Master Monk Fan-su on the Nature of The Flow, Dragons: the Thaumic Devourers, and any book in the 523 class in the library - which pertain to fundamental particles.