Background
The story begins when the lawyer Gabriel John Utterson hears from his cousin Richard Enfield of an ambiguous, solitary, violent man called Hyde. This Hyde is said to have "trampled" over a girl whom he met on the road, leaving her bruised and terrified; whereupon Enfield ordered him, backed by several other people, to pay a fine to the girl's family. Hearing this tale, Utterson is perturbed; a friend of his, Dr Henry Jekyll, has made a will declaring that in the event of the doctor's death or disappearance, Hyde should inherit all his property. Suspecting trouble, Utterson seeks to investigate Hyde.
This investigation begins as a matter of curiosity and concern despite Dr Jekyll's assurances that Hyde is nothing to worry about. That changes when Hyde is seen committing a savage murder of a respected Member of Parliament, Sir Danvers Carew. As Utterson assists in the investigation of the crime, Jekyll becomes more and more reclusive and sombre. This leads Utterson to believe that Hyde has some influence over Jekyll, which he is using to conceal himself.
Eventually, Jekyll isolates himself in his laboratory gripped with an emotional burden that no one can comprehend. Another friend of Utterson's, Dr. Hastie Lanyon, suddenly dies of a horrific emotional shock with which Jekyll seems to be connected. Eventually, Jekyll's butler comes to Utterson to ask for his help to deal with a stranger who has somehow entered the locked lab and killed Jekyll. Together they discover that the stranger in the lab is Hyde, and they break in only to find Hyde dead by his own hand and Jekyll nowhere to be found.
Eventually, Utterson reads three letters left for him from his deceased friends. The first one is a will made out to his name. The second is from Lanyon and reveals that he witnessed firsthand that Hyde is none other than Jekyll physically transformed into the other identity by means of a potion of Jekyll's design.
The other letter is a confession from Jekyll which reveals what occurred when he realised that every man has two aspects within him – good and evil – which constantly wage war upon him. Acting on the theory that it was possible to polarise and separate these two aspects, he created a potion that could change a man into an embodiment of his evil side, thereby also making pure his good side. After using the potion on himself, Jekyll became physically smaller as his evil nature became predominant; this persona was called Edward Hyde. The potion did not work as planned, in that the shape-changing was successful, but the identity of Jekyll remained unchanged while adding an alternate character who was purely evil. After a few trial runs as Hyde, Jekyll soon began to undergo the change regularly in order to indulge in all the forbidden pleasures that he would never commit otherwise. However, the Hyde aspect himself began to grow strong beyond Jekyll's ability to control it with a counter-agent. Eventually, Jekyll wakes up in bed one day to discover that he has turned into Hyde overnight. He resolves to give up Hyde for good, but the allure proves too strong to resist, and after two months he takes the potion once more.
This time, Hyde does not just indulge himself; he commits murder, and can no longer be seen in public for fear of being recognised and sent to the gallows. This reassures Jekyll, and he attempts to redeem himself for the actions of Hyde by being charitable. However, as a result of vainglorious thought, once more he undergoes the transformation, without the aid of his potion, in a park in broad daylight. He manages to avoid capture by finding a hotel room. He writes to Lanyon, asking him to fetch from his study the drawer in which the counter-agent is found.
Lanyon complies, and Hyde shows up at his house unrecognised. He takes the potion, as although he has begun to despise Jekyll, he fears recognition and the resulting death even more. He changes into Jekyll before Lanyon's astonished eyes. Heartbroken by this shocking revelation, Lanyon wastes away and dies.
Jekyll finds that he can now only remain in his original form with the potion in his system. Eventually Jekyll ran out of the unique components to the potion, and in particular a "salt" of which he had initially acquired quite a large quantity. New supplies of this salt did not produce an effective potion, which he initially attributed to an impurity in the new supplies, but finally concluded that it was the initial order that was impure, and that an "unknown impurity" in it was vital to its effectiveness. As he had no way of acquiring any more of this impure salt, he was doomed to remain as Hyde permanently.
In the end, Jekyll decided to write the confession letter, and he finally "dies" as he transforms completely into Hyde. Hyde commits suicide, through poison, when Utterson and Jekyll's butler try to force their way into the laboratory.

Analysis
There have been dozens of major stage and film adaptations, and countless references in popular culture. The very phrase "Jekyll and Hyde" has become shorthand to mean wild or controversial and polar behaviour. Most adaptations of the work omit the reader-identification figure of Utterson, instead telling the story from Jekyll and Hyde's viewpoint, thus eliminating the mystery aspect of the tale about who Hyde is; indeed there have been no major adaptations to date that stay close to Stevenson's original work, almost all introducing some form of romantic element.
For a complete list of derivative works see "Derivative works of Robert Louis Stevenson (by Richard Dury). There have been over 123 film versions, not including stage, radio etc. This is not an inclusive list, it is major and notable adaptations listed in chronological order:

1887, stage play, opened in Boston. Thomas Russell Sullivan's Dr. Jekyll and Mr. Hyde. This was the first serious theatrical rendering, it went on to tour Britain and ran for 20 years. It became forever linked with Richard Mansfield's performance, who continued playing the part up until 1907. Sullivan re-worked the plot to center around a domestic love interest.
1912, movie USA, Dr. Jekyll and Mr. Hyde (1912 film). Thanhouser Company Directed by Lucius Henderson starring James Cruze amd Florence Labadie
1920, movie USA, Dr. Jekyll and Mr. Hyde (1920 film). Directed by John S. Robertson. The most famous of the silent film versions, starring an inspired John Barrymore in a bravura performance. Plot follows the Sullivan version of 1887, with some elements from The Picture of Dorian Gray.
1920, movie Germany, Der Januskopf (literally, "The Janus-Head," Janus being a Roman God usually depicted with two faces). Directed by F.W. Murnau. An unauthorized version of Stevenson's story, disguised by changing the names to Dr Warren and Mr O'Connor. (Murnau more famously filmed an unauthorized version of Dracula in 1922's Nosferatu.) The dual roles were essayed by Conrad Veidt. The film is now lost, which is especially frustrating given that the good doctor's butler was played by none other than Bela Lugosi!
1931, movie USA, Dr. Jekyll and Mr. Hyde (1931 film). Directed by Rouben Mamoulian. Widely viewed as the classic film version, known for its skilled acting, powerful visual symbolism, and innovative special effects. Follows the Sullivan plot. Fredric March won the Academy Award for his deft portrayal and the technical secret of the amazing transformation scenes wasn't revealed until after the director's death decades later. "This is when JEEK-ull (IPA: [ˈdʒiːkəl]) became JEK-ull (IPA: [ˈdʒɛkəl]), the movie pronunciation."
1941, movie USA, Dr. Jekyll and Mr. Hyde (1941 film). Directed by Victor Fleming. Largely an imitation of the 1931 movie, it stars Spencer Tracy, Ingrid Bergman, and Lana Turner.
1955, TV USA, Hyde and Hare. Directed by Friz Freleng. Bugs Bunny Meets Dr. Jekyll And Mr. Hyde.
1960, movie UK, The Two Faces of Dr. Jekyll (released in the US as House of Fright and Jekyll's Inferno). Directed by Terence Fisher. A lurid love triangle and explicit scenes of snakes, opium dens, rape, murder and bodies crashing through glass roofs. Notable in that an aged and ineffectual Dr Jekyll becomes handsome and virile (but evil) Mr Hyde.
1963, movie USA, The Nutty Professor. Directed by Jerry Lewis. This screwball comedy retains a thin plot connection to the original work. Its enduring popularity has given it a significant role in the cultural visibility of the Jekyll and Hyde motif. Lewis re-works the Victorian polarised identity theme to the mid-20th century American dilemma of masculinity.
1968, TV USA/Canada, "Robert Louis Stevenson's The Strange Case of Dr. Jekyll and Mr. Hyde". Starring Jack Palance, directed by Charles Jarrott and produced by Dan Curtis of Dark Shadows fame. Shown in two-parts on CBC in Canada and as one two hour movie on ABC in the USA. Nominated for several Emmy awards, it follows Hyde on a series of sexual conquests and hack and slash murders, finally shot by "Devlin" (as Utterson is renamed).
1968, Music USA/England, The Who's John Entwistle released a song called Dr Jekyll and Mr. Hyde. The song followed a the plot of the story but also had a strange, dark sense of humor.
1971, movie UK, Dr. Jekyll and Sister Hyde. Directed by Roy Ward Baker, starring Ralph Bates as Jekyll and Martine Beswick as Hyde. The earliest work to show Jekyll transform into a woman. Recasts Jekyll as Jack the Ripper, who uses Sister Hyde as a convenient disguise to carry out his murders.
1971, movie UK, I, Monster. Directed by Stephen Weeks, starring Christopher Lee in the Jekyll/Hyde role and Peter Cushing as Utterson. Recasts Jekyll (with a name change to Dr Marlowe/Mr Blake) as a 1906 Freudian psychotherapist. Retains a fair amount of Stevenson's original plot and dialogue.
1972, movie Spain, Dr. Jekyll y el Hombre Lobo, a Paul Naschy film in his long-running series pits Dr. Jekyll against a werewolf.
1973, movie USA, directed by David Winters. A musical version made for television with original music by Lionel Bart starring Kirk Douglas as Jekyll/Hyde, with co-stars Michael Redgrave as Danvers, Stanley Holloway as Poole and Donald Pleasance as Fred Smudge. Nominated for Emmy award (Outstanding Achievement in Music Direction of a Variety, Musical or Dramatic Program - Irwin Kostal {music director})
1981, TV UK, with David Hemmings in the dual role and directed by Alastair Reid. This version gave a twist to the usual ending when the body turns into Mr Hyde upon his death.
1989, movie USA, Edge Of Sanity, a low-budget remake with Anthony Perkins as a Jekyll whose experiments with synthetic cocaine transform him into Hyde, who is also Jack The Ripper.
1989, NES Video game.
1989, TV UK, with Laura Dern and Anthony Andrews in the dual role. This version, adapted by Babylon 5 creator J. Michael Straczynski, was similar to Hammer's 1960 version in that Mr Hyde is the more physically attractive of the two; Dr Jekyll is depicted as a shy, mousy asocial scientist & Hyde is a handsome sociopath.
1991, Stage play, opened in London. Written by David Edgar for the Royal Shakespeare Company. The play is notable for its fidelity to the book's plot, though it invents a sister for Jekyll.
1995, movie USA, Dr. Jekyll and Ms. Hyde. Starring Tim Daly and Sean Young. Daly plays a perfumist who inherits the notebooks of his great-grandfather, Dr. Henry Jekyll, and performs experiments to refine the formula, but after adding estrogen to the mix, he turns into a ruthless nymphomaniac (Young) determined to climb the corporate ladder.
1996, movie USA, Mary Reilly. Directed by Stephen Frears. Starring Julia Roberts and John Malkovich and based on the 1990 novel Mary Reilly by Valerie Martin, a re-working of Stevenson's plot centered around a maid in Jekyll's household named Mary Reilly.
2007, TV serial UK, a 6 part BBC serial named Jekyll aired on June 16th 2007 starring James Nesbitt as Tom Jackman, a modern Jekyll whose Hyde wreaks havoc amongst modern day London. Jackman's transformation into Hyde is triggered by an as yet unknown cause, but it is not from a potion as it is in the book. Hyde is also credited as a completely new personality rather than the dark side of Jackman, with Hyde calling Jackman 'daddy'. The serial is not an adaptation of Stevenson's book, but rather a continuation taking place in the present day: the original book and both Dr Jekyll and Stevenson are prominently featured within the story.