An eternal fire:  For whatever reason, Heraclitus believed that the universe was an eternal fire (see Fire).  The universe did not come into existence, nor will it ever go out of existence.  Even though he believed in God/gods as did Homer and Hesiod, he did not think the gods created the universe.  The universe remains in its logical ordering, with all of its changes being regulated measure for measure.

"Changing, it rests" (B84):  It is unclear from the fragment itself what Heraclitus claims rests by changing, but he most likely was referring to the universe.  Heraclitus used many analogies and word pictures to explain his views; two pictures seem helpful here.  A fire is constantly changing as it rises up to the heavens.  The flame blazes up and up and never at any moment does it seem constant ... and yet we call it the same fire.  In fact, if the fire stopped changing in this way, it would die down and stop existing all together.  It is dependent on its change for its existence.  The same can be seen by looking at a river.  Heraclitus said something similar to this: "As they step into the same rivers, different and different waters flow" (B12).  The river is constantly changing as the water flows; at any given moment, the water at one location will be different than it was the previous moment.  And yet we call it the same river.  In fact, if the water stopped flowing, the river would not longer exist.  Like fire, the river is dependent upon change for its constant existence.  This helps us understand B124 about the barley drink; it also seems to be analogous to the whole universe.

Flux:  Underneath the ordered cosmos, closer inspection reveals a constant struggle of opposites and a state of rapid change.  The river as a whole rests and stays constant, but the story is quite different for individual water molecules.  Heraclitus' view of flux strongly influenced Plato (this view of constant flux led Plato to claim that knowledge about the world was impossible -- one could only have knowledge about the constant, unchanging forms).

Cosmic Cycle:  Heraclitus' elements changed in an ordered cycle.  Modern scholarship has placed a large emphasis on the details of this change, though I think the emphasis is not in Heraclitus himself.  As a metaphysician, the details of the cosmic changes were not as important to him as the fact that the cosmos always changed and in an orderly fashion.  Charles Kahn has a good discussion about the history of the debate on the cosmic cycle (Heraclitus 147-153).
    Heraclitus believed in at least three elements: fire, water, and earth.  B76 mentions air, but some scholars reject it as spurious.  Kahn has a good discussion on B76 and the scholarly debate about air (Heraclitus155-156).  What is clear about the cosmic cycle is that fire turns into other things, and other things turn into fire.  This happens 'measure by measure,' so that at every moment there is the same amount of fire, earth, water, &c.  (See Robinson 98-101 for a different interpretation.)
    The Stoics believed in ecpyrosis, or 'conflagaration': fire, they said,  would eventually consume all other elements.  They attributed this view to Heraclitus, and it is impossible from the fragments to discern if Heraclitus held such a view.  I tend to think that his views about an ordered cosmos and stabilitiy-in-change are inconsistent with ecpyrosis, but many earlier scholars do not think so.  These fragments given above under Cosmic Cycle include the so-called evidence for ecpyrosis in Heraclitus.

Sun:  The best account of Heraclitus' view comes from Diogenes Laertius.  Apparently, Heraclitus believed that the sun is a giant bowl orbiting the earth.  It does not always have something inside the bowl.  Every day, water from the sea would turn into fire (as part of the cosmic cycle, most likely as an 'exhalation') and fill the bowl.  By the time the bowl rose above the horizon, it would be blazing with fire.  The reverse would happen when the sun set: the fire in the bowl would turn into water and join the sea.  The empty bowl (it seems) would travel under the earth to prepare for another trip around the earth.  In this way, the sun is new every day.

Moon:  The moon works just like the sun, with only one key difference.  The bowl of the moon spins on an axis as it orbits the earth.  This creates the different phases.  When the bowl is faced completely away from us, the moon would be new; when it is faced towards us, the moon would be full.