Serenity is an incarnation of the Moon Goddess Serene, and Queen of Silver Millennium.
Her main duties were to look after the Silver Crystal and to raise her daughter, Princess Serenity, into a suitable heir. At that time there was a war between the Earth and the Moon as the people of Earth grew jealous of the long, rich lives that were lived by those on the Moon Kingdom. Despite this conflict, the Earth's Prince Endymion decided to fight for the Moon (partly because he was in love with Princess Serenity, and partly because he didn't approve of what his people were doing)
Unfortunately, Queen Metallia (the ruler of the Dark Kingdom) attacked the Moon, destroying everything. Everyone was killed except for Queen Serenity and her advisors, Luna and Artemis. Mourning the loss of her daughter as well as all her subjects, Serenity used the Silver Crystal to seal away Queen Metallia and to send everyone who had died that day to be reborn in the future. Sadly, Queen Serenity had used the full power of the Silver Crystal, meaning that she died immediately after using the crystal. Her spirit then remained on the Moon and watched over the Earth until Princess Serenity was born again.
Queen Serenity has a very small role in the anime. Towards the end of the first series she appears as a sprite to the Sailor Senshi, revealing the past of the Moon Kingdom to them. This sprite appears again to Usagi at the beginning of the R series and returns the Silver Crystal to her along with a new transformation brooch. After this she is never seen again.
In the manga, Queen Serenity's role is slightly larger: instead of appearing as a sprite to the Senshi, there is a holographic image (though she looks exactly the same). She also appears after Usagi defeats Queen Metallia and asks her to take her place as Queen of the Moon (but Usagi refuses). The story becomes more interesting in the Super S manga, where Usagi has a series of flashbacks to when she was a child in the Moon Kingdom. It is revealed that the evil Queen Neherenia was the sister of Queen Serenity, and that both originally came from somewhere else in the galaxy before they arrived at the Moon. When Princess Serenity was born, Neherenia tried to convince Queen Serenity to join her in her evil plans. When Serenity refused, Neherenia put a curse on the baby Princess and tried to claim her victory, but was soon defeated when Queen Serenity sealed her in a mirror.
When the princess had reached four or five years old, she started having nightmares, predicting the outcome of her curse. It is here that Queen Serenity gives Usagi the most important piece of advice that she ever receives in the entire series: "We all have a star in our heart. Darkness and light are always right next to each other. Show a little fear or a drop of tear and the darkness will grow and start to attack, consuming the light. Serenity, you must always keep the star in your heart shining strong to defeat the dark and evil souls. That is your most important mission, Serenity..."