In the play's final scene, Romeo finds Juliet's "dead" body and, rather than face life without her, swallows a vial of poison moments before Juliet wakes up. When Juliet realizes her husband is dead, she tries to kill herself by kissing Romeo. Since there's not enough poison left on his lips, she stabs herself with Romeo's "happy dagger" (5.3.3). (Sexual allusion intended.)
No spoilers to worry about here: as tragic as it is, the ending of Romeo and Juliet shouldn't surprise anyone. We're told from the get-go that our "star-crossed lovers [will] take their life" (Prologue). We also know that Romeo and Juliet belongs to the genre of "tragedy," and Shakespeare's tragedies always, always, always end in death. (You can read more about this by going to "Genre.) The point of reading or watching Romeo and Juliet isn't to find out what happens, but to watch it happen—and to feel some strong emotions along the way.
In the Prologue, the Chorus also tells us that "their death [will] bury their parents' strife," and it does. In one of the most ironic moments of the play, the couple's parents are so devastated by the deaths of their children that they kiss and make up, each father promising to erect an elaborate statue to commemorate the other's child (5.3.3).
Hmm. Do we detect a bit of competition here? When Montague announces his plans to "raise [Juliet's] statue in pure gold," he basically tells Capulet he's going to outdo him. "But I can give thee more," he brags (5.3.3). How long do you think this peace is going to last?
If you want to get all metaphorical, there's also a way of seeing the end as the ultimate sexual fulfillment. Shakespeare scholar Marjorie Garber points out that the "cup" Romeo drinks poison from is a traditional symbol of female sexuality. Juliet, in contrast, stabs herself with Romeo's dagger—a traditional image of male sexuality. (Do we have to explain these symbols? Think genitalia.) Garber argues that, symbolically, Romeo and Juliet combine physical death and sexual climax (source).
This reading makes sense. Death and sex are linked throughout the entire play (which you can read more about in "Symbols") and Juliet does say that ingesting poison by kissing Romeo's lips would "make [her] die with a restorative" (5.3.2). In other words, she's suggesting that the kiss and the poison would heal or "restore" her by reuniting her with her husband. But, since poison isn't a viable option for her, she chooses to unsheathe Romeo's sword and then thrusts it into her own body.
(We didn't even try to put sexual innuendo in that sentence.)
After you read the ending of Shakespeare's play, check out the ending of Franco Zeffirelli's 1968 film version of the play. If you can snag a copy, check out the ending of Baz Luhrmann's, too—we can't find a clip online, but it's definitely worth Netflixing.