The Milesian school was a group of Presocratic Greek philosophers in the city of Miletus in Ionia, who flourished from 585 BCE (when Thales was reported to have predicted a solar eclipse) to about 525, with the death of Anaximenes. It's commonly thought that Thales was a teacher of Anaximander, who in turn taught Anaximenes; these three together make up the Milesian school. They are also sometimes grouped together with other philosophers from Ionia -- collectively, and predictably, they're referred to as the Ionian school. But this sort of classification is somewhat misleading because of the vast differences between the first Ionians and those that came later on. Where the Milesians were primarily concerned with a philosophy of nature and determining what the material stuff of the cosmos was, later Ionians like Heraclitus had very different aims, often more focused on reason and thought than on nature. Thus the Milesians deserve to be treated as a discrete group, on their own terms.
Thales was the first philosopher to advance a materially monistic view of the cosmos, which came to be shared by all three of the Milesian philosophers: that is, the idea that all things are derived from a single substance, that all things are related in that they are comprised of the same (single) thing, "the substance persisting but changing in its attributes" (Presocratics Reader, 10, no. 1). He proposed that this 'first principle' was water, though according to Aristotle's commentary it is unclear as to why he chose water over other things -- perhaps it was because the received mythical cosmology from the East has to do with life as we know it springing out of primordial oceans (as in the Mesopotamian creation account, the Enuma Elish), or because Greece was surrounded by water and the earth itself was thought to rest upon it, or because "the nourishment of all things is moist" and thus moisture itself has to be that from which all things originate (10, no. 2). But regardless of Thales' motivation or logic in choosing water as his first principle, this conception of the unity of the material stuff of the cosmos was an important jumping-off point for later philosophers.
Thales also proposed that "all things are full of gods", which can be taken to mean that "[the soul] is mixed in the whole [universe]" (11, no. 4). Aristotle gives an account of Thales stating that magnets must possess souls, since they contain within them the power to move things; that "the soul [itself] was something that produces motion" (11, no. 5). We can interpret these two together as a way of leaving motive force to the realm of the supernatural: though Thales' account of the cosmos is more scientific than his predecessors, the poets, this aspect of his thought is under-developed by comparison, as it provides no similar account of what exactly causes movement.
A second potential problem with Thales is that he does not explain what it is that allows the sensible world to manifest itself to us as it does: what makes some things look different from other things, what accounts for discrepancies in taste and smell and feel between sensible objects, and indeed how it seems evident (to the senses at least) that water itself is somehow different from things that are not-water. This is perhaps the most serious problem with Thales' particular brand of material monism: after choosing as his first principle something that has definite characteristics of its own, he neglects to account for the processes that allow it to change its characteristics -- to change it from water into air and fire and stones and trees and all the other things that make up the world. All things might be water, but not all things appear to us to be water.
Anaximander, following after Thales, appropriated the concept of material monism, but in his case it was slightly more sophisticated in its application. He said that all things were not water, but instead something infinite and indeterminate -- something he called the apeiron. Apeiron, like water for Thales, "does not have a first principle, but [...] seems to be the first principle of the rest"; it "is divine", and it "[contains] all things and [steers] all things" (12, no. 7). Straight away this shows an influence from Thales: although with that said, Anaximander comes straight out and says that the apeiron itself is divine, instead of implying it and leaving it to supposition as Thales did (by saying "all things are full of gods" without elaboration, leaving us to assume that since all things are water, water itself probably partakes in divinity on some level). It differs, though, in that the apeiron does not have any defining characteristics of its own, the way that water does: it is a first principle which resembles nothing that comes out of it. Anaximander makes no reference to the movement of the apeiron; perhaps in its eternal nature it is static, providing direction and impetus for things coming to be out of it and passing away into it but remaining itself fundamentally unchanged.
The apeiron shows an improvement (or at least a development) upon Thales' first principle precisely because of its boundless and indeterminate nature. This undoes the difficulty presented by having a determinate first principle, with sensible characteristics of its own. The apeiron is indefinite, so it can manifest itself as anything perceptible by the senses -- because anything perceptible has to be different from it already, merely by virtue of its being perceptible. The objection to water as a first principle that has already been raised falls down when applied to Anaximander's apeiron.
The 'scientific' (anachronistically speaking) explanation of the universe put forward by Anaximander is more complete than that of Thales, because it provides a description of how opposites like 'hot' and 'cold' came to be, and an attempt to account for the existence of the celestial bodies that doesn't rely on the machinations of the gods as the poets depicted them. So far as hot and cold are concerned, something capable of producing them "was separated off [from the apeiron] at the coming to be of this cosmos", and the hot element rose up to surround the cold, wreathed about the earth (12, no. 8). The resulting "sphere of flame" splintered off into additional spheres that came to rest at various heights above the earth (which itself is at rest), creating the sun and the moon and the stars (12-13, nos. 8-9). Clearly this is more complex and involved than Thales' cosmology, which involved only the earth floating on the water that gave birth to it with no reference to the celestial.
Material monism was the central tenet of Anaximenes' system, too, just as it was for Thales and Anaximander. But where Anaximander had taken a step forward in removing the constraints of definite characteristics for his first principle, Anaximenes seems to take a step backward: his first principle is "not indeterminate as Anaximander held, but definite [...] it is air" (14, no. 16).
At first glance this is more akin to Thales' original proposition than it is to Anaximander's development upon it, because the first principle is back to being something with characteristics of its own. But Anaximenes' position is more advanced than both of those, because in a certain sense it manages to combine them: air as a first principle is both definite and boundless. Definite, in that air certainly has its own characteristics -- we can breathe it, when the wind blows we can feel it on our skin and observe as it rustles the leaves of trees; boundless, in that it is a substance neither solid nor liquid that defies being gathered up and contained. And it is divine: "it comes to be and is without measure, infinite and always in motion" (15, no. 19), in agreement with Thales' definition of the soul as that which causes movement, inherent to the whole universe.
Anaximenes' description of the earth's place in the cosmos relies on an example to demonstrate its rationality: like Thales, he has the earth floating upon his first principle (air, of course, instead of water) as a consequence of its origin. In this case, since "when the air is felted the earth is the first thing to come into being, and it is very flat", it floats upon the air -- it doesn't cut through it, but instead "covers it like a lid, as bodies with flatness apparently do", and as a result stays stationary (16, no. 23) -- just as in Anaximander's cosmology, but for more justifiable reasons.
Perhaps most importantly, though, Anaximenes takes the important step of explaining precisely how it is that the first principle can come to be made manifest as things other than itself -- in this case, through the processes of condensation and rarefaction.
Becoming finer [air] comes to be fire; being condensed it comes to be wind then cloud, and when still further condensed it becomes water, then earth, then stones, and the rest comes to be out of these (14, no. 16).
Like Anaximander, Anaximenes has his first principle give rise to hot and cold, not as substances in their own right but as properties that other substances might have. This, too, takes place as a result of air's becoming condensed or rarefied: "matter which is contracted and condensed is cold, whereas what is fine and 'loose' [...] is hot" (15, no. 21). He gives the example of an exhaled breath made quickly through taut lips being cool, and more slowly through relaxed lips being warm, but the same could just as easily be said of water (condensed air) being cooler than fire (air in its most rarefied form). This account of the origin of hot and cold is more useful than Anaximander's because it gives us an example to work with alongside the idea, and this shows an important distinction between the two: though Anaximenes' account is still mostly hypothetical, it is no longer strictly so, as he gives examples to clarify what he meant and demonstrate its application in the sensible world.
It seems clear that in this way there is some sort of progress made from Thales to Anaximenes; at the very least, all of these are real efforts to conceive of the world in way vastly different from the poets, approaching the question of what the universe is from an angle that is more scientific than mythological in its methodology. But for all that, the Milesians' work still seems incomplete -- none of them pay much attention to things outside the observable natural world. That direction would be taken up by Presocratics who came later: namely, the Eleatics, Heraclitus, the Pluralists, and the Pythagoreans.
Page references from A Presocratics Reader, Patricia Curd, ed. Indianapolis: Hackett Publishing Company, 1995.
Some background from The Presocratic Philosophers, 2nd ed. by Kirk, Raven, & Schofield (Cambridge: Cambridge University Press, 1999); and From the Origins to Socrates, 4th ed. by Giovanni Reale, trans. John R. Catan (Albany: State University of New York Press, 1987).