Shakespeare’s style

The most striking feature of Shakespeare is his command of language. It is all the more astounding when one not only considers Shakespeare's sparse formal education but the curriculum of the day. There were no dictionaries; the first such lexical work for speakers of English was compiled by schoolmaster Robert Cawdrey as A Table Alphabeticall in 1604. Although certain grammatical treatises were published in Shakespeare's day, organized grammar texts would not appear until the 1700s. Shakespeare as a youth would have no more systematically studied his own language than any educated man of the period.
Despite this, Shakespeare is credited by the Oxford English Dictionary with the introduction of nearly 3,000 words into the language. His vocabulary, as culled from his works, numbers upward of 17,000 words (quadruple that of an average, well-educated conversationalist in the language). In the words of Louis Marder, "Shakespeare was so facile in employing words that he was able to use over 7,000 of them—more than occur in the whole King James version of the Bible—only once and never again."
Shakespeare's English, in spite of the calamitous cries of high school students everywhere, is only one linguistic generation removed from that which we speak today. Although the Elizabethan dialect differs slightly from Modern English, the principles are generally the same. There are some (present day) anomalies with prepositional usage and verb agreement, and certainly a number of Shakespeare's words have shifted meanings or dropped, with age, from the present vocabulary. Word order, as the language shifted from Middle to Early Modern English, was still a bit more flexible, and Shakespeare wrote dramatic poetry, not standard prose, which gave some greater license in expression. However, Elizabethan remains a sibling of our own tongue, and hence, accessible.
This facility with language, and the art with which he employed its usage, is why Shakespeare is as relevant today as he was in his own time.
Introduction
In the England of Shakespeare's time, English was a lot more flexible as a language. In addition, Shakespeare was writing as a dramatic poet and playwright, not as a scholar or historian. Combine the flux of early modern English with Shakespeare's artistic license (and don't forget to throw in a lot of words that have either shifted meaning or disappeared from the lexicon entirely), and there are some subtle difficulties in interpreting Shakespeare's meaning some 400 years after the fact. As with most popular playwrights of any era, Shakespeare uses language with facility and power, but with a colloquial freedom as well.
That having been said, there are a number of ways to unlock Shakespeare's meaning based on habit and context. The highlights—based on importance and frequency—are presented below.
The Elizabethans loved the sound of their language on stage. Even the most inept of early plays rhymed and alliterated (click for an example), and it was Marlowe's "mighty line*" rather than his sensational plots that made his plays so popular. There was an energetic debate about the appropriateness of English adopting more difficult words from other languages; Shakespeare himself used many unusual words, but at the same time he satirized the excessive use of elaborate language.
In grammar, the sound of the language was more important than the logic of the way a sentence was constructed. If it suited their purpose to invert natural word order or to repeat words for emphasis, writers would do so.
In Elizabethan grammar there were fewer transitional words and other grammatical signals, and so it is sometimes difficult to distinguish subordinate clauses from main clauses or subjects from objects, especially in verse, where the writer might shorten words or leave them out altogether* if the metre demanded it. (More about special Elizabethan words like "thou."*)
Superlatively dead
In Shakespeare's time the comparative adjective "more" and the superlative adjective "most" were used as intensifiers, as "very" would be today. As a result, the plays are full of what we would call double comparatives and double superlatives*. When Hamlet describes the deceased Polonius as "most still, most secret and most grave" (3.4.214), he is using this kind of expression, rather than describing Polonius as the most dead of all dead people.
A similar effect was achieved by doubling words so that the first use of the word intensified the second. Hamlet in particular contains several of these uses: "Excellent, excellent well" (2.2.173), "quite quite down" (3.1.154) and of course the opening of one of his most famous soliloquies, "O that this too too sullied* flesh would melt" (1.2.129).
Shakespeare is renowned for his puns*, particularly in his comedies. English had only recently become the language in which the well-educated expressed themselves*. It was "fire-new" (a metaphor taken from the art of the blacksmith) and offered writers a wonderful opportunity to play with words.
In Love's Labour's Lost, Don Armado is a knight who loves words. He is writing to the young King of Navarre reporting a law-breaker:
Great deputy, the welkin's vice-regent, and sole dominator of nature. . . So it is, beseiged with sable-coloured melancholy, I did commend the black-oppressing humour to the most wholesome physic of thy health-giving air. . . Where . . . I did encounter that obscene and most preposterous event, that draweth from my snow-white pen the ebon-coloured ink, which here thou viewest, beholdest, surveyest, or see'st.(1.1.218-43)
Elizabethan pronunciation differed significantly from our own. Vowels were in the process of changing, in a process known as a "vowel shift"--the same process that has given us so many different accents today. Thus there are a number of words that would have made perfect rhymes that now sound like half-rhymes: "love" and "prove," for example.
In Henry IV, Part One, Falstaff tells Hal, seemingly inexplicably, "If reasons were as plentiful as blackberries, I would give no man a reason upon compulsion." There is a pun here, but the modern audience would be hard-pressed to notice it, unless "reason" were pronounced in the Elizabethan manner, which would sound something like "raisin." The pun then becomes obvious, and the line makes much more sense. (Click to listen* to this passage.)
In Julius Caesar, Cassius puns on "Rome" and "room"-- and again the words were pronounced alike. (Click to listen* to this passage.)
Andrew Gurr reviews scholarly work on Shakespeare's pronunciation in this useful article in the electronic journal Early Modern Literary Studies See the Renaissance Faire for Proper Elizabethan Accents.
If the spectators in Shakespeare's theatre were so delighted with the gorgeousness of language, why did Shakespeare sometimes write in prose? Sometimes, as in Henry IV, Part One, there seems a clear division between the formal verse of the court and the rambunctious prose of the tavern. . .
But there are as many exceptions to this neat division as there are instances when it is accurate. Hamlet's powerful speech "What a piece of work is a man" (2.2.303-16) is delivered entirely in prose, and what are we to make of a play like The Merchant of Venice, in which the prosaic, business-obsessed characters of Venice usually speak in verse and the poetic, imaginative characters of Belmont often speak in prose?
One consistent difference seems to be that verse is used when there are passages of high feeling and increased intensity, while prose is often the language of wit and play.
The rhythm of prose
Shakespeare's prose is as masterly as his verse, and often even more dense with meaning (check the footnotes of a passage of prose, and see if there are fewer than for verse). Prose is the vehicle for many of Shakespeare's wittiest characters: Falstaff, Beatrice, Rosalind, the Porter in Macbeth, Autolycus in The Winter's Tale, and many others.
While most of the memorable passages from the plays are in verse, many memorable scenes are in prose, or a mixture of the two: prose is one of the many ways in which Shakespeare keeps the rhythmic counterpoint* of his language alive.