How do characters in Romeo and Juliet play specific roles in moving the story forward? 

There are several characters that make up the supporting
cast that help move the plot forward in Romeo and Juliet.  I'll
discuss a few here:

Benvolio is a sincere, good natured
young man, who being Romeo's cousin and close friend serves as Romeo's confidant who
hears out his problems, which often have to do with Romeo's love interests.  Benvolio's
advice is always upbeat and encourages Romeo to move foreward in his life and not get
overly depressed over a girl.  In his most significant piece of advice, he tells Romeo
to attend the Capulet masquerade party in order to meet other women.  Romeo is hesitant
and pesismistic about the venture, but he agrees.  This becomes essential to the plot
because this is where Romeo meets Juliet and their whirlwind romance
begins.

Lord Capulet appears to be a sober
and understanding father, who cares deeply about his daughter and is interested in her
feedback regarding her future desires, unlike many other fathers during that era. 
However, Tybalt's death greatly affects his rational and progressive thinking, and as
the head of the household, he feels it is his duty to take immediate action in order to
stop the grieving throughout the Capulet family.  His rush to judgement forces Juliet
into marrying Paris, putting Juliet in a predicament that leads her to fake her death,
pushing the plot on toward the bloody final action and
resoultion.

Tybal plays the role of Romeo and the rest of
the Montagues' main antagonist.  Tybalt sees himself as an honorable young man with the
greatest loyalty to his family and, therefore, maintains the greatest hate toward all
Montagues.  His constant acts to defend his family's honor and to commit violence
against the Montagues continue to fuel the feud despite the efforts of the Prince, Lord
Capulet, and others' attempts to quell it.  His insistence to fight Romeo in Act III is
at least partially responsible for his fight with Mercutio that proves to be fatal for
the latter.  This prompts Romeo to give in and fight Tybalt, which ends in Tybalt's
death and Romeo's banishment from Verona.  The killings and banishment put an end to the
Friar's hopes that Romeo and Juliet's union can bring peace to the Montague and Capulet
families with no further bloodshed.  It also puts Romeo and Juliet in a situation where
they are facing life apart from one another, which allow them to consider the most
desperate means to be together.