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At the very beginning, Shakespeare suggests that his sonnet is magnificent by using very magnificent comparisons in lines 1-2:
Not marble, nor the gilded monuments ,Of princes, shall outlive this powerful rhyme;
In contrast, he uses the word "rhyme" at the end of line 2, which is often used to signify common and mediocre, even bad, poetry, which suggests that it is the subject of his sonnet that lends magnificence to the verses.
This is only confirmed in lines 3-4:
But you shall shine more bright in these contentsThan unswept stone, besmeared with sluttish time.
Shakespeare comments that his subject will be brighter in his sonnet than an old and dirty stone, again suggesting, by equating his poem with dirt, that his sonnet does not live up to the subject. He likewise calls Time "sluttish", clearly comparing it unfavourably to his female subject. Also, the reference to stone recalls the structures alluded to in line 1.
Lines 5-6 (a new stanza) begins a new idea:
When wasteful war shall statues overturnAnd broils root out the work of masonry,
Shakespeare has so far spoken of two destructive forces: time and war. He is here describing war destroying stone structures, which relates back to the "marble" and "gilded monuments" in line 1, that likewise do not last.
Lines 7-8 continue the war theme:
Nor Mars his sword nor war's quick fire shall burnThe living record of your memory.
These lines talk of more war, and how it shall not destroy the poem. "Mars his sword" is a possessive, using the his genitive. "Living" contrasts with the destruction of the non-living structures in lines 1 and 5-6, meaning that the subject lends not only magnificence, but a living soul to these verses.
The next stanza does not talk about survival, but of human appreciation. He continues to praise his subject:
'Gainst death and all-oblivious enmityShall you pace forth; your praise shall still find roomEven in the eyes of all posterityThat wear this world out to the ending doom.
There is still a suggestion of survival, but survival of human appreciation, and not of the verse itself. "Doom" refers to Judgement Day, suggesting in the context of the rest of the poem that this poetic record of his subject will survive, and be praised, to the end of time. The slight deviation of the meter in the words "Even in" creates emphasis for this permanency.
The ending couplet is a summary of the survival theme:
So 'till the judgment that yourself arise,You live in this, and dwell in lovers' eyes.
The couplet not only summarizes the rest of the sonnet, but also seems to contradict itself. "Judgement" goes with the talk of Judgement Day in the last stanza, and implies that the subject is alive and will be judged on that day, but "dwelling in lovers' eyes" suggests that the subject is love itself. Thus, Shakespeare seems to consider the subject so lovely that he is a personification of Love, which cannot be conquered and to which no poetry can do justice.