What is the meaning behind the concept of "gods" in Buddhism?

(I'm not claiming this to be the official teaching of any Buddhist school, but if you talk to enough Vajrayana practitioners you will see that this interpretation is a distillation of their common views.)

If you try to study anthropology or ethnology you can't help but notice that most cultures in the world have more or less the same notion variously called spirits, deities, demons, ghosts etc.

And in many of these cultures these deities/spirits are not just named characters but instead associate with and manifest a certain quality of human character or a certain abstract idea.

Claude Lévi-Strauss, Joseph Campbell, Carl Gustav Jung and many others studied these phenomena trying to find common patterns or structures. The consensus seems to be that these repeating themes and images represent certain elements inherent to human mind or collective psyche.

A deity can be described as a cyclic mental tendency, a thoughtform - being made up of, and influencing, the thoughts and motivations of its carriers. This symbiotic relationship between a deity and its carriers can be compared to the more recent secular concepts of the corporation (as a juridical person or legal entity), the consumer brand, and the meme or memeplex. You can read more about memes in the works of Richard Dawkins and Susan Blackmore.

A deity is a personification of a stable self-supporting non-physical tendency inhabiting the noosphere - the sphere of information, causation, and potential latencies.

This is how deities are understood in Vajrayana, as manifestations of the (non-dual) inter-human mind-realm.

Pali & Sanskrit have the word "indriya", which in Pali can mean "controlling principal" or "directive force".

In Sanskrit or Vedic language, "indriya" can mean "fit for or belonging to or agreeable to Indra". In the Pali suttas, Indra is the king of the gods. DN 23 says Indra is to whom appeal should be made when needing protection.

Therefore, it all valid religions (which excludes fake religions, such as Protestantism), the notion of "god" or "gods" appears to represent at least both a controlling power & protector/guardian.

For example, SN 11.5 about different types of gods as different types of rulers and Iti 106 refers to the parents of children as "gods".

Refer to my answer here for further perspectives.

To understand "gods" as used in Buddhism, I found this sutta helpful:

AN4.54:5.1: And how does a god live with a goddess? It’s when the husband … is ethical, of good character … And the wife is also … ethical, of good character … That’s how a god lives with a goddess.

From this I understood that the Buddhist gods embody stable, long-lasting, peaceful, harmonious, expansive and inclusive qualities. Remarkably, Buddhist gods are not immortal, but are also bound to change, to death, to suffering:

AN3.80:9.8: If Ānanda were to die while still not free of greed, he would rule as king of the gods for seven lifetimes, and as king of all India for seven lifetimes, because of the confidence of his heart.

Taken together, these two add up to a sobering conclusion. The perfections that we might wish for are all bound to change, to death, to suffering.

AN8.35:4.6: ‘If only, when my body breaks up, after death, I would be reborn in the company of the Gods of Brahmā’s Host!’

AN8.35:4.7: They settle on that idea, concentrate on it and develop it. As they’ve settled for less and not developed further, their idea leads to rebirth there.

And here we understand that the Buddha is telling us that we will need to go further. To end suffering, we will need to gently put aside our wish for heavenly perfections and follow the Noble Eightfold Path to realize the deeper Four Noble Truths.

In his commentary of MN 49, Ven. Thanissaro wrote:

It is a play on the word brahma, which is not only a noun denoting the highest levels of devas, but also an adjective meaning "of great or high power."

MN 49 is entitled the "Brahma invitation", so Ven. Thanissaro says that it means firstly an invitation by the Brahma, but it also means (through a word play) an invitation of great power or high power, which is an invitation by the Buddha to Mara (the personification of the temptation of sensual enjoyments and becoming) to refute the Buddha's claim that he has put an end to becoming once and for all.

An allegorical use of the term "Brahma" can be seen elsewhere. For e.g. parents are like the "Brahma" to their children in Iti 106 and the Buddha himself is like the "Brahma" to his disciples in Iti 100, who are "born from his mouth".

In Iti 112, we see the statement "he set the Brahma-wheel going". What is Brahma wheel? Ven. Thanissaro comments that it refers to the Dhamma wheel of SN 56.11:

The Brahma-wheel = the Dhamma-wheel, the name of the Buddha's first sermon, so called because it contains a "wheel" that lists all twelve permutations of two sets of variables: the four noble truths — stress, its origination, its cessation, and the way leading to its cessation — and the three levels of knowledge appropriate to each truth: knowledge of the truth, knowledge of the task appropriate to the truth, and knowledge that the task has been completed. This wheel constitutes the Buddha's most central teaching.

Here, calling the Dhamma wheel "Brahma wheel" denotes that it is the Great Wheel.

The most interesting use of this phrase comes in Iti 68:

One whose passion, aversion, & ignorance
are washed away,
is said to be
composed in mind,
Brahma-become,
awakened, Tathagata,
one for whom fear & hostility
are past,
one who's abandoned
the All.

Here's the Pali version from here:

“Yassa rāgo ca doso ca,
avijjā ca virājitā;
Taṃ bhāvitattaññataraṃ,
brahmabhūtaṃ tathāgataṃ;
Buddhaṃ verabhayātītaṃ,
āhu sabbappahāyinan”ti.

So the Tathagata is awakened (buddham) and became Brahma (brahmabhūtaṃ).

In other words, he became great, like a god to us.

Returning back to MN 49, the opening verse states:

Now on that occasion an evil viewpoint had arisen to Baka-Brahma: 'This is constant. This is permanent. This is eternal. This is total. This is not subject to falling away — for this does not take birth, does not age, does not die, does not fall away, does not reappear. And there is no other, higher escape.'

Ven. Thanissaro's footnote reads:

Baka Brahma here appears to be referring both to his Brahma world and to the state of mind that enables one to inhabit his Brahma world.

So, this points to Brahma as referring to states of mind and being (or becoming - bhava) which are of great and high power.

The sutta shows that such a mind state and state of being is not permanent or eternal, and there are even higher states of mind and being.

So, we can say that this is the "subtle" interpretation of Brahma.