A dynamic character changes significantly throughout the story. A static character remains substantially the same. Romeo is a static character. His main personality traits are over-reaction, rashness, and emotional instability. When we first meet Romeo, we see two of the three. He is upset over the reject from Rosaline. As a result, he has sunk into a deep depression. Despite the calm counsel of his friend Benvolio, Romeo refuses to rise out of his depression. He refuses to accept that there could possibly ever be another woman for him.
Enter Juliet. Romeo falls quickly in love. He rushes into her garden, despite it being the home of his enemy and dangerous. He pledges himself to Juliet after one quick meeting. He arranges to marry her. The Friar himself is shocked by Romeo's quick change of heart: "Holy Saint Francis! What a change this is! Is Rosaline, who you did love so dearly, So soon forsaken? young men's love, then, lies Not truly in their hearts, but in their eyes."
Romeo does marry Juliet, but his new love and new position as a husband does not change him. Although he has sworn not to fight, he quickly changes tunes after Mercutio is killed and kills Tybalt. Then, when news comes to him that Juliet has been killed, he acts rashly and without logic. Rather than seeking the Friar for more information, Romeo kills himself. He never learned his lesson.
Romeo is a dynamic character. He changes through the course of the play, and markedly. The most obvious evidence of this is his shift in love objects, and how that changes how he acts. Romeo longed for Rosaline, and thought his love was true…until he met Juliet. Then he couldn't play at love anymore, but was transformed by it. He still tasks risks, but the risks he used to take for fun, he now takes to see his beloved. He's willing to kill, to flee the city, anything, to be with her. That's dynamic.