What are the significant characters in The Ramayana?

There are many significant characters in The Ramayana.  
Whenever addressing a sacred text vital to a spiritual tradition, there will be many important characters.  This is especially so with The Ramayana, as a case could be made that every character mentioned is important in their own way.  This is to emphasize that what follows is by no means comprehensive. 
Lord Rama is probably the most important character in The Ramayana.  He is an avatar of Lord Vishnu, and his narrative forms the basis of the text. Born to King Dasaratha, he is the example of dharma, or adherence to the overarching structure of the universe.  Lord Rama carries himself as the perfect son, prince, warrior, husband, and brother.  His defeat over the demon-king Ravana establishes how human beings must live their lives with duty and honor.
Sita is Lord Rama's wife.  Born to King Janaka of Mithila, she is an avatar of Goddess Lakshmi.  As Lord Rama's wife, Sita devi demonstrates an equal commitment to dharma.  As beautifully honorable as Lord Rama is as a husband, she is as much an equal as a wife.  She is kidnapped by the demon-king Ravana, setting in motion the confrontation between her husband and her tormentor.  While in Lanka as Ravana's captive, Sita devi demonstrates her strict virtue in refusing to acquiesce even in the slightest to Ravana's gestures. Sita is the perfect daughter, wife, sister, daughter-in-law, and sister-in-law.
Lakshmana is one of Lord Rama's brothers.  Along with Shatrughna and Bharata, Lakshmana is very devoted to his brother.  Lakshmana is the avatar of Adi Sesha, the reclining serpent upon which Lord Vishnu rests.  When Rama is exiled to the forest for fourteen years, Lakshmana feels it his duty to accompany him.  He is a vigilant protector of Rama and Sita.  When Sita is kidnapped, Lakshmana accompanies Rama to confront and eventually kill the demon-king of Lanka.
Ravana is the antagonist of The Ramayana.  While Rama represents structure, honor, and adherence to a structure that envelops human beings, Ravana represents self-indulgence and excess.  Ravana is the demon-king of Lanka. Through extreme penance to Lord Brahma and Lord Shiva, he acquired boons of incredible power. He used this power to consolidate his control over the universe.  Rama is incarnated from Lord Vishnu in order to defeat the existential threat that Ravana posed to justice and goodness.  Ravana kidnaps Sita as a way to showcase his power because he covets her beauty.  His kingdom in Lanka is the very embodiment of ostentatiousness and sensory gratification.  Ravana has many wives, to which he wishes to add Sita.  He is defeated by Lord Rama at the end of the epic, representing justice's triumph over injustice.
Hanuman is the commander of the monkey army that pledges to serve Lord Rama in his quest to retrieve Sita. Lord Hanuman was born to Vayu, god of the winds, and Anjani.  As a child, Lord Hanuman was incredibly brilliant, but also very mischievous.  His lack of focus ceased the moment he met Lord Rama. Hanuman immediately devoted himself to Lord Rama, and was willing to do anything and everything for his guru.  
Hanuman's devotion to Lord Rama is an important part of The Ramayana.  Through his immense strength and skill, he is able to jump across the ocean to enter Lanka, and collect vital intelligence that he relays to Lord Rama. He is the first one to find Sita devi, reminding her not to lose faith in Lord Rama.  Hanuman orders the army of monkeys, the Vanaras, to build a bridge to Lanka made of floating stones that allows Lord Rama, Lakshmana, and the army to enter Lanka and challenge Ravana.  When Lakshmana is struck down by Ravana's potent arrows, Hanuman saves him by bringing an entire mountain to Lanka that contains a precious herb.  Hanuman's selflessness is the example that human beings are meant to follow.
These are the primary characters in The Ramayana. There are many more that play vital roles in the development of the narrative.  In temples all over the world, shrines are built to Lord Rama, Devi Sita, Lakshmana, and Hanuman in acknowledgement of their efforts in overcoming the injustice that Ravana represented.