This poem is often compared to Hamlet's "To be, or not to be" soliloquey in that it expresses the speaker's alienation and lonlieness and confusion. In fact, Prufrock refers to this in the poem:
No! I am not Prince Hamlet, nor was meant to be;
In this poem, the speaker is carrying on the same type of self dialogue and analysis. He is overwhelmed by his situation, the plight of the so-called modern man who is full of doubts about the meaning of life and his place in it. He continually questions himself, expresses doubts (do I dare? and do I dare?), and finds himself socially alienated and incapable of the human contact that he so desires.
He not only expresses these self-doubts about his present situation, but also fears the future -
I grow old, I grow old
I shall wear the bottoms of my trousers rolled
The pessimism of modern society permeates this poem - the same pessimism Eliot expresses in most of his works -- look at some of the titles: The Waste Land, The Hollow Men. Even this poem is just the opposite of a "lovesong".
There is so much more to say about it, but this will get you started. Read the excellent analysis of the theme of modernism here on eNotes.
The above post is quite good, still I would like to add a few lines.
Thomas Stearns Eliot's Prufrock is a perfect representative of the modern human.
Yes, the speaker in "The Love Song of J. Alfred Prufrock" is much in the "to be or not to be" position. But, he considers his state worse than that of Hamlet. He thinks, Hamlet could at least succeed in fulfilling his aim at the end. But he is more like Polonius - coward, passive, and spineless - who is more comfortable with the oblivion: "Am an attendant lord, .../ Full of high sentence, but a bit obtuse/ At times, indeed, almost ridiculous- / Almost, at times, the Fool".
Prufrock, in this modern world, feels very secluded, isolated, and alienated, not fit for this world at all. He can not adjust himself with the society around. Modern people seem to be pretentious, and their boring, endless arguments on nonsense topics make him more frustrated: "Streets that follow like a tedious argument/ Of insidious intent.../ Oh, do not ask, "What is it?"/Let us go and make our visit". Women around him are just fake, though, they show off a lot: "In the room the women come and go/ Talking about Michelangelo".
He gets more depressed like Tiresias in The Waste land, since immoral acts have enveloped the modern world. There is hardly truth and beauty in relationships; there prevails mostly perverted sexuality. The images like "yellow smoke" and the metaphorical reference to cat in the 3rd stanza stands for the perverted sexuality and lust.
He gets confused under such a circumstance where there are pretentiousness, dishonesty, immorality and lack of love. He scares to propose the lady he loves because of the fear of being rejected. He knows that, appearance is everything in today's world, the reality does not matter that much. So, he is concerned about his look which is just unimpressive: "With a bald spot in the middle of my hair-/ They will say.../ Do I dare/ Disturb the universe?". Thus, he feels more lonely.
Prufrock is exhausted because of the feeling of nothingness in his life. He does not find any meaning of his existence. He has no individual identity in this modern world: "I have measured out my life with coffee spoons". That is why, he feels more comfortable while he is in oblivion. But, the reality does not allow him to escape, and at the end, he has to wake up and endure the pain: "We have lingered in the chambers of the sea/ By sea-girls wreathed with seaweed red and brown/ Till human voices wake us, and we drown".
Eliot's Prufrock, being a modern man who is far more romantic and anti-modern, clearly defines the vital problems of the modern age. And, thus, the poem becomes a wonderful instance of those few literary works which directly deal with the modern life.