a kind of life long after her body is gone, because of this little poem. It's a pretty powerful sentiment, is it not?

Not only does the speaker insist that his beloved is more beautiful and perfect than a summer's day—already a massive claim on its own—but he argues, persuasively, that she will never "dim" or "decline" as a result of natural changes. This speaks to one of humankind's greatest fears: that we will die and be forgotten. Shakespeare's speaker promises his lover that this will never happen to her, making the poem almost universally appealing.

Firstly, Shakespeare's "Sonnet 18" is so famous because, outside of perhaps "to be or not to be," it has some of Shakespeare's most famous lines and phrases that most people have heard even without knowing the poem. For example "Shall I compare thee to a summer's day?" and "darling buds of May."

Secondly, is his answer to the question he poses in the first line of whether she deserves to be compared to a summer's day. His answer seems to be does summer deserve to be compared to the woman. Sometimes summer is too hot or windy. Other times the sun disappears behind the clouds. This woman, however, is always beautiful and will remain beautiful for the remainder of her life.

He immortalizes the woman through simple and clear but powerful language and metaphors that anybody can relate to. Summer, for example, isn't constant or reliable. As he states:

Sometime too hot the eye of heaven shines
And often is his gold complexion dimm'd

Aside from the answer you have already received, the poem is beautiful.  It is a love poem which trips effortlessly off the tongue, and every single person on earth can relate to this subject.  Love is always a popular topic for poetry and music. The speaker attempts to convince the one he loves to what extent he loves her...

Perhaps the most romantic lines in the poem are:

But thy eternal summer shall not fade

Nor lose possession of that fair thou ow’st;

Nor shall Death brag thou wander’st in his shade,

When in eternal lines to time thou grow’st;

So long as men can breathe or eyes can see,

So long lives this, and this gives life to thee.

Where the speaker basically says that her beauty will never fade, nor shall death claim her and their love since both will be eternally alive and youthful in the lines of this poem that many eyes will read even after they have physically left this world. 

Good question; I have a couple of possibilities. First of all, it is perfect "textbook" example of a Shakespearean sonnet--the sonnet form that uses three quatrains of iambic pentameter followed by a concluding couplet.

Secondly, the entended comparison of the poem (a young boy and a "summer's day") uses vivid imagery and unforgettable diction. Most people who are not poetry lovers will still recognize and remember the poem's first two lines:

"Shall I compare thee to a summer's day? / Thou art more lovely and more temperate . . . "

This particular sonnet has endured well for these reasons. It has become a part of our culture, from movie references in films like Dead Poets' Society to spawning a band called the "Darling Buds."

How many people do not remember a glorious summer day that perfectly matched their inner mood?  Sonnet 18 recalls this memory to people.  And, is not this perfection of love not an appropriate way to describe one's sincere and lasting love by saying it is better than a summer's day? "Thou art more lovely and more temperate" is the quintessential compliment.

SONNET 18
http://www.shakespeare-online.com/sonnets/18detail.html

This sonnet is a love poem, like so many of Shakespeare's sonnets.  In this one, the person to whom it is addressed is described in terms as beautiful as a summer's day.  She is lovelier, actually, since the wind of summer's days rough the darling buds of May and the sun is too hot sometimes.  Her beauty will not ever fade...not even in death...since as long as the lines Shakespeare are read she will live also.

Sonnet 18, Shakespeare's best known sonnet, describes beauty and decay, and references nature and someone beloved, making a comparison between both. What's significant about this sonnet is the self-reference -- although beauty decays as a person gets older and dies or as the seasons change from spring to winter, the couplet suggests that as long as this sonnet continues to be read, the beauty it describes will endure, like an eternal summer. (see Robert Frost's "Noting Gold Can Stay" for a more recent poem describing something similar.)

Shakespeare's "Sonnet 18" is suitable for an oral recital because the underlying metaphor comparing the eternal summer of beauty to the eternal summer of beauty in a written sonnet is at once familiar and yet unusual; therefore while it can be followed in an oral recitation because of its familiarity, it is also thought provoking for its unusual aspect. Another reason it is suitable is that there are not interconnected strings of word play as in "Sonnet 35" in which a single word play string comprises four lines; thus this sonnet will be easier than some to follow and attend to, as well as being easier to adequately recite. Another reason is that the turns at lines 5 and 9, turns (or voltas) that are definitive of sonnet structure and descend from Petrarchan sonnets, are clear as they lead into a change in subject within the sonnet's topic; thus in an oral recitation, the turns can be dramatized and emphasized. A final reason is that "Sonnet 18" is one of Shakespeare's more elegantly beautiful sonnet's as he isn't lamenting or scolding or trying to forgive someone stubbornly not interested in forgiveness.