As others have said, the letter doesn't get to Romeo because the messenger Friar John was not able to send the letter to Romeo (who is in Mantua) or enter Mantua himself due to the precautions taken by the town to prevent illness, i.e. bubonic plague (also referred to as "the Black Death"). Upon finding out that the letter cannot reach Romeo, Friar Laurence plans to reach Juliet before Romeo arrives. However, this plan fails as well; Romeo gets there first and upon discovering (thinking) that Juliet has died, kills himself.
An interesting side note is that Shakespeare wrote this play around the time that the plague was taking over Europe. In fact, due to extremely high death tolls, the theaters had closed down until the infection lessened. Upon reopening, Romeo and Juliet was one of the earliest plays to be performed at theaters. So the existence of the plague in the play must have resonated very closely with the audience that had just been through it themselves.