Hamlet's appeal over the centuries is clear. We can all relate to Hamlet in his angst and unaccepting nature that his mother had been married to his uncle in such a short time. His thirst for vengeance throughout the play is also intriguing, in both character analysis and the story itself. Hamlet serves as a tragic example of how a young and brilliant man can destroy himself in the quest for revenge, as he pulls down many people along the way.
The most notable individual is Ophelia, who loved Hamlet from the very start. But Polonius and Laertes opposed this kind of relationship, which serves to drive Ophelia over the edge. Hamlet denies his love for her, driving Ophelia closer to the brink, especially how she should "be sent to a nunnery." It is the death of Polonius by Hamlet's hand that does her in. She drowns herself in a brook, a testament to how high the cost of revenge is.
Even to those who do not know Shakespeare, the image of Hamlet holding up a human skull is a popular image with the Bard. The skull that Hamlet holds belonged to Yorick, a jester from Hamlet's childhood. Some people think Shakespeare wanted Elizabethan audiences to connect Yorick with Richard Tarlton, who was famous prior to Shakespeare and was dead by the time Yorick was mentioned.
The Prince of Denmark has been portrayed by many people. Laurence Olivier is one famous example in the British 1948 production of Hamlet. In most recent years, Mel Gibson and Kenneth Branagh have also played their hand as the title character.
Franco Zefirelli's 1990 Hamlet with Gibson is pretty much an action flick with most of the dialogue left out. I saw this version quite a while ago with "Lethal Hamlet" intense and brawny. Branagh's 1996 version, which I have not seen, is four hours long with all dialogue included. A 2000 version featuring Ethan Hawke as the brooding Hamlet was released in the style of Romeo + Juliet, in that it takes place in a modern setting.
Also famous is Hamlet's "To be, or not to be" speech. Those who have read into the play understand it is the Prince of Denmark's contemplation of suicide. This speech has been used many times and it also has been spoofed several times.
This video below comes from 1996's Hamlet, with Branagh as Hamlet uttering the famous soliloquy.