What does the line "Look on my works ye mighty and despair" from "Ozymandias" mean?

"Look on my works, ye Mighty, and despair!"

Basically, this line has two meanings. The literal meaning is "Look at all my great accomplishments, and envy me, because even if you think of yourself as mighty, you’ll never be as great as I am!" The subtle, ironic meaning is: "Look at how all of my great accomplishments have crumbled away into nothingness, and feel sad and hopeless, because even if you’re mighty, the same thing will happen to you after you’ve died!"

The line we’re talking about appears in the poem "Ozymandias," specifically near the end, at the climax (the most exciting part).

Let’s recall what’s going on in that poem. The speaker of the poem reports a story he heard from a "traveller" (line 1). This "traveller" remembers what he saw while visiting a desert in "an antique land" (line 1). It’s the broken and "shattered" (line 4) statue of some king named Ozymandias. The statue’s pedestal bears an inscription, which we read in lines 10–11:

"My name is Ozymandias, king of kings:

Look on my works, ye Mighty, and despair!"

The inscription is telling everyone to look at his "works," whatever they are—maybe glorious buildings, or maybe more statues. But here’s the twist. There’s nothing to look at. There are no "works" in sight, just the "boundless and bare" sands of the desert (line 13). Even the statue itself is a decaying "wreck" (line 13).

As you can see, the words the king says in line 11 are ambiguous: that is, we can interpret them in two different ways, both of which are valid. The poet, Percy Bysshe Shelley, most likely rendered this line ambiguous on purpose. It makes the poem intriguing. It makes the reader think, "Wait—I’m sure the great king, the one who called himself 'king of kings' (line 10) just meant to boast and brag when he talked about his works, and he made sure that the sculptor carved those boastful words into his statue…but now, so much time has gone by that his words have a completely different meaning. A meaning that the king didn’t intend at all! That's ironic."

Even more than ironic, line 11 is chilling. It suggests to readers that, after our inevitable deaths, and after the inevitable passage of time, not only will our accomplishments fade away into oblivion, but even the meaning of our words can warp. We die. Our "works" die. Even our words die. The poem, chiefly through line 11, implies that this bitter truth applies to even the mighty, even a king, even a "king of kings."

What is the meaning of these lines from Shelley's "Ozymandias"? "Look on my works, ye mighty, and despair! Nothing beside remains. Round the decay..."

It's lines from Shelley's sonnet Ozymandias, and to make sense of it you need to put them in context. Here's the whole poem, with your lines in bold:

I met a traveller from an antique land
Who said: Two vast and trunkless legs of stone
Stand in the desert. Near them on the sand,
Half sunk, a shatter'd visage lies, whose frown
And wrinkled lip and sneer of cold command
Tell that its sculptor well those passions read
Which yet survive, stamp'd on these lifeless things,
The hand that mock'd them and the heart that fed.
And on the pedestal these words appear:
"My name is Ozymandias, king of kings:
Look on my works, ye Mighty, and despair!"
Nothing beside remains: round the decay
Of that colossal wreck, boundless and bare,
The lone and level sands stretch far away.

So, your first line is part of the words engraved on a pedestal, boasting about the great works of the king Ozymandias. But the boast is ironic, because the works have been destroyed by time: only two huge fight and the shattered face of what once must have been a statue remain. "Nothing beside remains", our narrator tells us - there's nothing else left.

And, round the decay of that huge wrecked statue, all there is is sand - metaphorically representing the sands of time, which has brought the mighty (in this sense, quite literally) to its knees.

What is the meaning of these lines from Shelley's "Ozymandias"? "Look on my works, ye mighty, and despair! Nothing beside remains. Round the decay..."

Shelley, in these lines, is pointing out how the pride, egotism, and powerful feeling of immortality possessed by this ruler has turned to dust, and how "nothing beside remains" of his majestic kingdom and royalty.  The monument to this leader sits in the middle of a desert, abandoned, forgotten, eroding away.  However, during the king's time period, he probably felt that he was immortal and all-powerful, and hence had his statue inscribed with the ominious "Look on my works, ye mighty, and despair!" logo that was meant to emphasize his power.

But now the traveller is looking on all that is left of the ruler's works, and doesn't feel despair; rather, he feels pity, or nothing, as all that is left of the king's kingdom is the desert, "boundless and bare/The lone and level sands stretch far away". 

What is the meaning of these lines from Shelley's "Ozymandias"? "Look on my works, ye mighty, and despair! Nothing beside remains. Round the decay..."

Ozymandias was a pharaoh, usually called Ramses II, or the Great. Like many pharoahs, he spent much of his life constructing monuments to himself. Massive temples at locations like Abu Simbel and the Ramesseum at Thebes were intended to project his power, both to his contemporaries and presumably to posterity. Shelley's poem relates a description by a fictional traveler who has encountered one of his massive statues, broken into pieces. The pride, majesty and arrogance of Ramses still come across despite the ruined condition of the find, but they seem ironic, surrounded as they are by desert. The final stroke of irony is the inscription. Surrounded by desert wastelands, his entreaty to "look on my works, ye mighty, and despair" only underscores his mortality and the fleeting nature of his greatness. 

Paraphrase the following: "And on the pedestal these words appear: 'My name is Ozymandias, king of kings: Look on my works, ye Mighty, and despair!'"

The context for these words is as follows: a traveller tells the story of finding a broken statue amid a vast empty desert of sand. It is the statue of ancient king who has a sneer on his face.

On the base on which this statue once stood, these words, which I will paraphrase, are carved: I am named Ozymandias. I am the greatest of all kings. Look around you, other mighty rulers. See the powerful kingdom I have built and feel no hope.

The words imply that no other ruler can ever expect to rival Ozymandias. He can crush any of them with his vast power. Therefore they should feel despair.

However, in the context of an empty desert today, the words have the opposite meaning. They are ironic. They now mean the following: look around, mighty rulers, and know you have no hope, for your kingdom also will be reduced to nothing.

Paraphrase the following: "And on the pedestal these words appear: 'My name is Ozymandias, king of kings: Look on my works, ye Mighty, and despair!'"

Percy Bysshe Shelley`s sonnet Ozymandias is written in a genre sometimes known as ‘ubi sunt` (Latin for: `where are they?’) that has a thematic focus on the transitory nature of worldly glories (another frequently used Latin phrase is ‘sic transit gloria mundi’ or ‘so pass the glories of the world’). The narrator is describing a large statue. The statue was originally placed an a large base, with the words 'My name is Ozymandias, king of kings:/Look on my works, ye Mighty, and despair!' inscribed on the base. In these lines, imagine that the statue is talking to the observer, pointing out that Ozymandias is so powerful and has caused such amazing works (like the statue) to be built that the observer, no matter how important or wealthy, should despair of rebelling against Ozymandias, emulating him, or equalling him. Of course, the statue is partially destroyed, located in a remote area, and Ozymandias forgotten, making the perspective of the poem`s reader far different from that of the imagined observer.