Both Morgoth and Sauron were Dark Lords. Did Tolkien express any opinion about which he considered to have the greatest power at the height of their careers (either in the books or any of the extended materials)?
Morgoth is clearly more powerful than Sauron. This was not just an opinion but rather hard fact stated by Tolkien himself.
Morgoth is not only the original Dark Lord (before Sauron assumed the title) he is also the main antagonist of Tolkien's entire universe. He created everything that is bad in Arda in the first place and is prophesied to return from his exile in the Final Battle, to once again wage a great war before his permanent defeat.
Sauron is stated to be Morgoth's greatest servant but in the end he is just that, a servant of a greater power, even if he later took over the mantle of Dark Lord.
He could not hope to match his master's power, certainly not at his peak. Morgoth grew gradually weaker which led to his eventual defeat. Sauron might have been closer to his master's power then, but that is just speculation.
There is no question that Morgoth at his greatest, in the Years of the Trees, was much more powerful than Sauron ever was. But to clarify Ron Meyers' answer, Tolkien more specifically said that Sauron at his greatest (the end of the Second Age) was greater than Morgoth at his weakest (the end of the First Age), and that the reason for this was that Sauron was more clever in the use of his power.
Morgoth spent his power in the 'brute force' method of attempting to control the very matter of Arda itself, e.g by creating the orcs, trolls, and dragons, raising the gigantic Iron Mountains, and changing the weather (the winters became colder in Beleriand as the power of Morgoth increased). By the end of the First Age Morgoth had spent so much of his power that he had become bound to his earthly form, unlike the other Valar, and could be physically harmed, as he was in the duel with Fingolfin.
Sauron, on the other hand, channelled his own power into charm and deception - hence the title "Sauron the Deceiver". He helped to forge the Rings of Power in order to bind to his command those he gave the rings to, he sent forth his will to sow fear and confusion in the minds of his enemies, and manipulated various peoples of men into serving him from the beginning of his rule. He also raised mountain ranges and altered the weather, but such means were not paramount to him.
Now, Morgoth had inadvertently laid part of the foundation for Sauron by creating the orcs and trolls, but Sauron still chose not to continue down that path and opted to focus on messing with people's minds instead.
In Morgoth's Ring, a book compiled of Tolkien's notes on Morgoth, by his son, it states in Tolkien's own words that Morgoth is "Incomparably greater than the power of Sauron, concentrated in the One Ring." Instead of pouring his malice into a ring, Morgoth poured his into the creatures and Arda itself. Another quote is, : "The whole of Middle-earth was Morgoth's Ring."
Darth is right in his reference...Tolkien did, in fact say (myths transformed), that Sauron became more powerful, but I do not believe that he meant that Sauron was inherently more powerful. What I took this to mean is that Morgoth (once the most powerful, after Illuvatar) expended his 'essence' in marring the earth. In other words, what Sauron invested into the corruption of a 5 cm ring of gold, Morgoth invested into the corruption of an entire planet, leaving his energy lesser. Also, Sauron was cagier, more of a weasel - instead of fighting at Morgoth's side in the war of wrath, he hid and lived to fight another day. I think that Sauron (as literary "true evil" will), served his master as long as it suited him. in being Morgoth's servant. I think he was able to avoid the 'diluting' of his spirit the way Morgoth (and the other Valar) did.
I think it is certainly possible for Sauron to have been more powerful than Morgoth at the end of the Silmarillion but in no way shape or form could he match Morgoth at the top of his strength.
Going off what Sebastian_H said, clearly explains that Sauron is simply a Maia. Though that is of course a very powerful being to be in Middle Earth, a Maia is still a Maia. In no way, shape, or form stronger than, firstly an Ainur, and secondly possibly the strongest of the Ainur.
"...among them Nine were of chief power and reverence; but one is removed from their number, and Eight remain, the Aratar, the High Ones of Arda: Manwe and Varda, Ulmo, Yavanna and Aule, Mandos, Nienna, and Orome. Though Manwe is their King and holds their allegiance under Eru, in majesty they are peers, surpassing beyond compare all others, whether of the Valar and the Maiar, or of any other order that Iluvatar has sent into Ea."
Note the distinction between the Aratar and the Valar. Melkor is greater in power than any single member of the Aratar, who are greater than the remaining Valar (ie Vana,Tulkas etc).
"With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree."
Sauron is a Maia. Not even a Vala. Melkor is a Vala, and toe to toe, can take down any single member of the Aratar.
"but he fell back into evil, for the bonds that Morgoth had laid upon him were very strong."
Melkor is obviously the far greater power, even at his perceived weakest. It still took a full on assault by the powers to ensure Melkor's end and cast him into the void.
Sauron seemed to have greater relative power primarily because the power of the elves was ebbing and the world was far lesser than it was when Sauron's boss roamed Middle Earth...so much so that the powers of the West sent "cast" equivalent wizards to contend with Sauron and his plans instead of just sending any of the Valar to dispatch of Sauron...would have saved innumerable lives. It was all seemingly part of a greater plan to test and temper man to stand and rule as the creator had planned.
Remember that Morgoth only realised he was diminished after he met with the Valar again. Keep in mind that Morgoth had Sauron in his fortress, and Gothmog too, but he never noticed his diminished strength around them. Plus Morgoth corrupted men to be like him, and we see they clearly have his traits.
What does this mean? Well, the Valar are suddenly standing in his way. He doesn't even try to fight them, nor was he initially willing to challenge Fingolfin. His pride - combined with an unhealthy dose of anger - can limit his powers and weaken him very much.
This means that if Sauron was truly capable of fighting and defeating Morgoth in a duel, Morgoth would tremble in fear and subdue him out of anger and fear. In my personal opinion, after finishing preparations for the dragon host to be able to fight, Morgoth was weaker than he had ever been. Remember that this was at the end of the First Age, so by now, Sauron was more powerful than Morgoth.
Illuvatar is god as in GOD type número UNO GOD. The Valar, of which Morgoth/Melkor was the greatest, were less than him in every aspect. I see them as similar to Greek style Olympians, though the Morgoth story is very Luciferian. They were almost invulnerable. The most notable exception being the several times noted Melkor fusing himself to middle earth creating his mortality through his use of power.
The Maia were assistants/children of the Valar and lesser power/they can and are occasionally killed but by great power at great effort. Sauron was greatest of the Maia, Olórin was the wisest (Gandalf). In Judeo-Christian terms these are arch angels. The Balrog are fallen ones. Each of the Istari (wizards) were Maiar. Tulkas was a special case - I have seen him listed as Valar and as Maia. Thor like had greater strength than Melkor.
Elves were the first children. They were basically the angels sent to prepare the world for the second children, men who are the heroes of the epic and were men.
To the elves much knowledge and power was given, particularly those born in Arda (mount Olympus).
The essential story line is as follows:
Melkor plants seeds of corruption in the first elves, then with Ungoliant (spider of pure evil) kills Finwe (father of Feanor, creator of Simarils, which are jewels of pure light) steals the Simarils and kills the Trees of Light, then flees to Angband, his fortress on earth.
With Valinor despondent over the loss of the trees, Feanor and his sons were incensed over Finwe and vow to not stop until the Simarils are recovered and Morgoth is captured.
They kill Teleri elves to take their ships so they can sail to earth to get to Melkor, which angers the Valar, thus elves get little to no help battling Melkor. Elves arrive on earth and defeat Melkor's forces in the first great battle but aren't strong enough to capture Angband, thus they lay siege to the mountain fortress.
Long time of settling and elven kingdom building happens, Melkor creates dragons and becomes stronger, men emerge. Battle of unnumbered tears ends elven siege. Melkor is the unmatched power of Middle-Earth, but not strong enough to wipe out all the elves at once. From this battle epic tales of men spring forth throughout history. Elves have opportunities to turn the tide but the vow of Feanor interferes in some way to block the path.
Beren and Luthien: The woodsmen named Beren falls in love with elven princess of Doriath. To win her fathers approval he, with her aid, and the aid of Huan the hound take a Simaril from the crown of Melkor. Beren in the end dies but Mandos can not bear Luthiens heart ache and allows Beren to return in exchange for her immortality. Hurin, man hero of battle of unnumbered tears, and savior of elves was captured by Melkor. His torture was to live to see how Melkor tormented his son Turin. Turin Turambar was Herculean hero, first to kill a dragon. But personal tragedies led to his suicide.
Turin, son of Tuor, who I believe was Hurin's brother, was met by the Valar Ulmo, and sent to hidden the city of Gondolin to defend the king and his daughter. While he built defenses Melkor conquers Dagorth forest and Novgorod with out Turin to defend it. Beren and Luthien establish hidden woodland kingdom and have children. Turin wins the heart of the princess of Gondolin and they marry and have a son, Eärendil. The nephew of the king is jealous and betrays the secret of Gondolin's location to Melkor. The 4th great battle is the fall of Gondolin. Turin and his family flees through a secret escape.
Eventually Eärendil and the daughter of Beren and Luthien, Elwing, both being of half elf, half man decent, is deemed a special marriage. They have two sons Elros and Elrond.
Eärendil destined to be greatest among elves urged by Ulmo to sail for Arda to ask for help from Valinor and elves that remained in Arda. Long journey sails, Arda hidden, many adventures, eventually finds and successfully gets help to come. A large host of elves return and the war of wrath begins. Great unleashing of dragon army my Melkor lead by Acagoglon the black dragon. Tides of battle turns when Eärendil wearing the simarill won by Beren returns to earth riding an eagle kills Acagoglon. Melkor is captured with a chain forged by Aule, wielded by Tulkas, and banished to the void of space ending first age.