In Jewish and Christian mythology, the rainbow is a sign from YHWH. It reminds mankind that he will never again flood the earth as in the time of Noah (Genesis 9:12-17):

And God said, “This is the sign of the covenant I am making between me and you and every living creature with you, a covenant for all generations to come: I have set my rainbow in the clouds, and it will be the sign of the covenant between me and the earth. Whenever I bring clouds over the earth and the rainbow appears in the clouds, I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become a flood to destroy all life. Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth.” So God said to Noah, “This is the sign of the covenant I have established between me and all life on the earth.”

I am going to take a somewhat broad definition of “interpret” here to include scientific understanding of rainbows, since that is what I am most familiar with, and for the same reason I will discuss only Europe with one notable exception. For a more cultural discussion, I recommend the two excellent books “The Rainbow: From Myth to Mathematics” by Carl B. Boyer, and “The Rainbow Bridge: Rainbows in Art, Myth, and Science” by Raymond L. Lee Jr and Alistair B. Fraser.

Rainbows have forced the attention of scientists and natural philosophers since at least the time of Aristotle. He considered it to be the sun’s image reflecting off of a cloud, with the colors arising from the light being “weakened” by this reflection, and the circular shape due to geometric considerations.

This theory prevailed mostly unchallenged until the early 14th century, by which time significant advances had been made in the science of optics. This enabled two philosophers, Theodoric of Freiberg, and Kamal al-Din al-Farisi, to simultaneously and independently make nearly identical scientific advances. They realized that the rainbow was best understood as the cumulative effect of the interaction of the sunlight with the individual water drops, and in one of the earliest examples of scientific experiment constructed a model rain droplet out of a thin, water-filled glass sphere to investigate the rainbow phenomenon. Each concluded that the rainbow was not caused by a single reflection by the drop, but rather a refraction-reflection-refraction sequence. Further, they developed a model of the secondary rainbow (colloquially the “double rainbow”) as a refraction-reflection-reflection-refraction event, and also gave explanations the reversal of its colors relative to the primary bow, the appearance of Alexander’s dark band between the two bows, and the brightness of the sky interior to the primary bow.

This model was essentially qualitatively correct, but did not explain the particular, quantitative value of the angular size of the rainbow. By tracing the paths of sunlight through the drop, Descartes noticed that much of it “clumped up” around a maximal angle of refraction. Using Snell’s law of refraction and his own methods of analytic geometry, Descartes computed this angle to good accuracy. To explain the colors, Descartes thought that light was made of particles, and the spinning of the individual light particle was related to its color and its spread from his calculated angle. Newton later discovered that the color of light was directly related to the degree by which it was refracted by water (and glass), and so had a different characteristic angle at which it tended to clump up when scattered by the rain drop.

While the Descartes/Newton theory became generally accepted, there remained some issues which it could not adequately explain – notably the supernumerary bows. The supernumerary bows are additional bands directly below the violet at the bottom of the primary bow, often appearing as alternating purple and green. (See the top of this rainbow. ) The solution to this came in the increasingly important wave nature of light. Thomas Young and later Richard Potter gave an explanation of this as due to interference: the different rays of sunlight would take slightly different paths in the drop but still end up near the same Cartesian angle, but the small differences in their path lengths would cause them to be in-phase or out-of-phase and so interfere, producing the supernumerary bows. This was refined to a significant degree by Lord Airy, who incorporated the full wave nature of light (i.e. including the effects of diffraction) to predict the specific location of the supernumeraries more accurately.

Finally, the essence of the modern theory of the rainbow is understood as the problem of the scattering of electromagnetic plane waves off of a dielectric sphere, for which the solution was formulated by Gustav Mie in 1908, which he actually used originally to explain another beautiful phenomenon: the striking colors of gold colloids.

... also check out previous responses in these threads

So what have people thought of rainbows throughout history?

What did people make of, or think of, rainbows in the middle ages?

How much did the Ancient Greeks believe their myths?

In Australia, people speak of the Rainbow Serpent, which was the creator. Depending exactly which version you talked about, it was not necessarily positive or negative, it just was what it was. For instance, in some cultures it was seen associated with the end of rain, generally seen as a bad thing (but might not be).

Perhaps /r/Anthropology is a better place for this question.

In relation to the biblical story, the Babylonian (and Palestinian) Talmud(s) think of rainbows as a fundamentally negative phenomenon. This was a result of the idea that a rainbow was god's way of asserting that he was being stopped by his earlier promise not to destroy the world. The rainbow implied that destruction was narrowly avoided due to god's forbearance and covenant. Thus, particularly righteous rabbis "did not see the rainbow during their lifetime," i.e. through their intercession and extra-ordinary merit, they were able to justify the non-destruction of the world, and no appeal to the covenant was needed.

See B.T. Ketubot 77b (a folktale where R. Joshua b. Levi sneaks into the afterlife after tricking the angle of death):

"Elijah heralded him [Rabbi Joshua the son of Levi] proclaiming. ‘Make room for the son of Levi, make room for the son of Levi’. As he proceeded on his way he found R. Simeon b. Yohai sitting on thirteen stools of gold. ‘Are you’, the latter asked him, ‘the son of Levi?’ — ‘Yes’, he replied. ‘Has a rainbow [the latter asked again] ever appeared in your lifetime?’ — ‘Yes’, he replied. ‘If that is so [the other said] you are not the son of Levi’. The fact, however, is that there was no such thing [in his lifetime], but he thought, ‘I must take no credit for myself’."

Edit: English

This is what Sahagun wrote in the Florentine Codex about what the Aztecs told him about rainbows.

Book 7, pg. 18

As if arched - bent and rounded - it thus appeared. Varicolored, many-hued was its appearance. The single colors which showed it [were] green, dark green, blue green, and black; and yellow and orange and tawny; then vermillion and ruby red; and [various shades of] blue, and dark green.

And they said that when it appeared it revealed, made evident to men, and denoted - whereby it was known, realized and seen - that it would not rain; would not rain hard - would not pour. But it would break up the clouds. It would dissipate, impede, and quiet the rain - the downpour which wet, soaked, and drenched one. If clouds piled up, if heavy clouds blackened [the sky] so that everywhere it was dark, it only dissipated [them]. Although it rained, no longer was the rain heavy; no longer did it increase. It only sprinkled; it drizzled; a few drops fell; a haze, a fog; a thin mist drifted and fell. Or at most it sprinkled; the sprinkling and the spray continued.

And they said - it was averred - [that] if it appeared over maguey plants, because of it the green [leaves] yellowed, turned, dried, reddened, and withered. And also they said [that] when it appeared many times, thereby it was evident [that] the rains were to end forthwith. They said: "Soon the masters of the rain will go; already the Tlalocs are about to leave."

Not an academic source, to be sure, but the Wikipedia entry for rainbows in mythology gives a quick overview -- starting from that, you could build your research --

http://en.wikipedia.org/wiki/Rainbows_in_mythology