The term "star-crossed" means "ill-fated," or destined to turn out miserably, even destined to die (Random House Dictionary). Thus, "star-crossed lovers" would be a pair of lovers who are destined for their romance to end in misery.
The first way that Shakespeare proves that Romeo and Juliet are "star-crossed lovers" is by having them meet purely by chance, or fate, at a party that Romeo and his friends were not supposed to be at. In fact, Romeo tries to refrain from joining his friends in crashing the Capulet ball. He even warns them that he believes some misfortune will come of crashing the ball, even warning that his own "untimely death" will be a result of "this night's revels" (I.iv.116-117). In having the couple meet through fate and also foreshadowing their pending doom, Shakespeare proves that the couple are "star-crossed lovers."
The second way that Shakespeare proves that Romeo and Juliet are "star-crossed lovers" is by having several unfortunate events happen the moment after they marry that cause them grief and even separate them. The couples' misfortunes even culminate in their deaths. The first unfortunate event that makes them a miserable couple is that Romeo is challenged and provoked to kill Tybalt who had just killed his best friend, which results in Romeo's banishment. Romeo's banishment separates the lovers and brings them emotional anguish. While Juliet is mourning the loss of Romeo through his banishment, her father makes the decision to marry her to Paris in order to distract her and cheer her up, thinking that she is mourning Tybalt's death. Of course, the threat of marrying Paris leads to her suicide threat and to her faked death. Juliet's faked death leads to Romeo's actual suicide and to her actual suicide as well. Hence, we see that Shakespeare depicts Romeo and Juliet as lovers fated to meet and also as lovers who are destined for their love to end in misery, even death.