Why does Hamlet describe his world as an "unweeded garden" in Act 1, Scene 2?

This is one of Shakespeare's most brilliant metaphors. Shakespeare is expressing his own view of humanity through one of his characters. Hamlet is saying that most of the people in the world are either rank or gross. There is a big difference in those two words. An unweeded garden that grows to seed can be an interesting object of study. In fact, it is a wonder that artists such as Monet and Van Gogh did not paint unweeded gardens. The things that are "rank" in such gardens are the kinds of plants that do not grow tall but choke the flower beds and encroach on the walks, helping to create a picture of abandonment, failure and desolation. Lawns that are not mowed will either become dead and brown or else, if they can get enough rain water, will become thick and ragged-looking. The tangled individual plants will appear to be fighting for a foothold on the soil and a place in the sun. This is how Shakespeare perceived the majority of humanity--fighting one another to stay alive and not being worth the effort. He expresses a poor opinion of people in many of his plays, including King Lear, Macbeth, Measure for Measure, and Timon of Athens. Perhaps he thought of people as being weeds and neglected plants in a garden because such plants give the impression of fighting for their lives--every man, and every weed, for himself.

In addition to the vegetation that is rank, there are some weeds that take advantage of the untended condition of the garden to grow to their full height and even develop ugly flowers. Such tall weeds are usually marvelously ugly and misshapen. Milkweeds are a good example of weeds that are "gross." And Shakespeare's Hamlet is thinking that many humans who stand out above the masses are remarkably ugly, or "gross." The best example of a gross human being in Hamlet is King Claudius. He is a villain, a liar, a drunkard, and a lecher, but he is a very interesting character. He stands out above the common herd of men who are "rank." Claudius is one of the outstanding individuals who can be described as "gross." Shakespeare's metaphors and similes are almost always simple, natural, unpretentious, and strikingly appropriate. 

An unweeded garden takes all the joy away from a garden that should be beautiful. Hamlet is still in mourning for his father's death, and what disturbs him most is that his mother has married his father's brother, as if she is not in mourning for his father any longer. In Act One, Scene Two, Hamlet's world is an unweeded garden at this moment:

It is an unweeded garden
That is going to seed, only things that are decaying and
Disgusting grow there. That it should come to this!
Only dead for two months!

Clearly, Hamlet cannot accept his mother marrying his Uncle Claudius. His heart is ripped in two. His world is full of decay and disgust as is a garden that is unweeded. There is no beauty in his world at the present. Not only has he lost his father, but he feels he has lost his mother. He feels that she should have mourned longer than two months. Well, actually, Hamlet is upset because it has been less than two months:

Only dead for two months! No, not so much, not two.
So excellent a king that, compared to this king, was
A magnificent man to a beast, so loving to my mother,
That he might not allow the winds of heaven
To blow on her face too roughly.

Hamlet compares his world to a decaying, unweeded garden. He cannot find beauty in his world because his father is dead and his mother is already married to his Uncle Claudius.

Why does Hamlet describe the world as "an unweeded garden"?

Hamlet is comparing humanity to the things growing in an unweeded garden that grows to seed. Such neglected gardens are commonly seen where houses have been standing vacant for a long time. Hamlet says that things rank and gross possess them merely.

Fie on't! ah, fie! 'tis an unweeded garden
That grows to seed; things rank and gross in nature
Possess it merely.

There is a distinct difference between "rank" and "gross." Ornamental vegetation that is rank is overgrown, choked, entangled. Lawns will become rank if they are not regularly mowed. They are an ugly sight. It is unusual for a poet to create ugly imagery as a metaphor to represent an ugly condition. The rank vegetation represents the ugliness that Hamlet sees in ordinary men. Growths in neglected gardens that can be called "gross" are not rank but stand out separately, towering over the rest. Milkweeds are a common sight in neglected gardens. They are tall, ugly, misshapen, grotesque. They seem to flaunt their ugliness. They bear flowers, but the flowers are colorless, ugly and randomly placed on the crooked stalks. Hamlet might have been thinking of King Claudius as one of the "gross" specimens of humanity.

Most people would probably not stop and spend time looking at an unweeded garden that grows to seed. They are depressing spectacles, but Shakespeare obviously appreciated them as lessons from reality and as potential metaphors and similes. He was not solely interested in pretty sights or in creating aesthetically pleasing imagery. He was more interested in expressing truth than in flattering nature. Some of his characters are beautiful examples of ugly people, including Iago, Caliban, Edmund, Regan, and Polonius. These characters are "gross." Characters like Rosencrantz and Guildenstern might be considered "rank." There are two of them but they seem like Siamese twins.