How does love influence the characters in Romeo and Juliet?

Love is a powerful force in this play that drives characters out of their comfort zone.
For example, if they had not fallen passionately in love, both Romeo and Juliet would have stayed within the protective bubble of their feuding families, never questioning that their own family was in the right and the rival family in the wrong. Falling in love with someone from "the wrong side of the tracks" forces each of them to reevaluate everything they formerly believed.
Juliet, for example, manages to look beyond Romeo's Montague last name to see his true worth, declaring "a rose by any other name would smell as sweet." In other words, the Montague name might stink to her, but Romeo is wonderful. This shows that love motivates her to think for herself and, hence, to be more mature. Love also motivates her to deeply grieve the death of her cousin Tybalt and yet to remain loyal to Romeo, his murderer and now her husband. Love transcends hate and family history.
Finally, the love that unites Romeo and Juliet and leads to their suicides has a profound influence on the Montague and Capulet families. Both families loved their now-dead child, and both families reconcile due to their children's lost love, ending the bitter feud.

Love is what drives most of what Romeo and Juliet do. When they fall in love, they are willing to sacrifice a great deal just so that they can be together. However, romantic love is not the only form of love which dominates this play. Familial love also plays a major part—even competing with romantic love. When Juliet learns that Romeo has killed her cousin Tybalt, she first curses Romeo as a villain; but then her love for him overcomes this anger. Her love for Romeo also motivates her to overcome her fear over drinking the sleeping potion (so that she can fake her own death).
However, love also inspires characters to do ignoble things. Romeo's friendship with Mercutio is what causes him to violently avenge his friend's death. Additionally, both Romeo and Juliet commit suicide when they see (or in Romeo's case, thinks) their lover has died. So, in Romeo and Juliet, love is a double-edged sword.