Irony is the contrast or conflict between what's stated explicitly and what's really meant. In other words, it's the difference between our understanding of a situation and what actually happens.
Here are some of the forms of irony in Romeo & Juliet:
Verbal irony. The words literally state the opposite of the writer's (or speaker's) true meaning. For example, the Prologue in Act I opens with "Two households, both alike in dignity, . . ." When you first read this, you may think that the two families are pretty dignified or honorable. As the play goes on, however, you realize that each family is violently competitive. They are similarly undignified.
Situational irony. Events turn out the opposite of what was expected. What the characters and audience think ought to happen isn't what eventually happens. In Shakespeare's play, the young lovers do end up spending eternity together, but not in the way the audience had hoped.
Dramatic irony (sometimes called tragic irony). Facts or events are unknown to a character in a play or story but known to you or other characters in the work. For example, the audience knows that Juliet took a sleeping potion and isn't really dead. Romeo's suicide affects the audience even more because of this knowledge.
Other forms of irony that you can find in literature include Socratic irony, comic irony, Roman irony, and cosmic irony (fate). By the way, you can pronounce it EYE ruh nee or EAR uh nee.