Chibiusa first appears in the second season, Sailormoon R, as a mysterious pink-haired child that falls from the sky to interrupt a romantic date for Usagi and Mamoru. She arrives demanding the ginzuishou, brandishing a gun. When she fires it at Usagi, Mamoru panics until he realizes the gun only shoots darts - already symbolizing she may not be as threatening as originally thought. Within a flash the girl is gone, leaving the couple puzzled...but when they find her later on being targeted by a new enemy, all they can do is try to save her.
When Tuxedo Kamen holds her unconscious in his arms for the first time, he suddenly gets a vision of what looks to be a large city, crystalline in nature. Astonished at what just happens, he looks down at the mysterious girl. Surely there must be more to all of this than they thought. As it turns out, Chibiusa is Usagi and Mamoru's daughter from 30th Century Crystal Tokyo where they rule as Neo-Queen Serenity and King Endymion.
Chibiusa takes to Mamoru very much right away, calling him "Mamo-chan" just like Usagi does. Turns out Mamoru is very good with children and seems to love spending time with Chibiusa. Once we are aware that she is his daughter, it's easy to see some personality she has inherited from each of her parents. She acts very much like Usagi, but also is extremely clever and caring, and mature for her age (specifically in SuperS).
During the end battle in Sailormoon S, Chibiusa's pure heart has been removed from her body by Mistress 9 and her life hangs in the balance as the sailor senshi fight. It's here when one of Mamoru's natural powers shines - he holds Chibiusa on his lap and holds her hands. A glowing aura appears around her - Mamoru uses his energy to keep Chibiusa alive until an image of Sailorsaturn appears to return the pure heart.
Chibiusa brings out a side of Mamoru that we hadn't previously seen in the series - he tries to cheer her up with a Tuxedo Kamen puppet, he seems to know just how to talk to her, he helps her with her homework, gives her piggy-back rides, so on and so forth. Chibiusa also keeps Usagi and Mamoru in constant touch with one another during their breakup.
The Greater Love
I find this a little absurd but on a list of reasons to hate Mamoru, many people claim that he loves Chibiusa more than Usagi. I don't find this to be true at all. It's obvious that he loves them both greatly and equally, however he loves them in different ways. Usagi is his soulmate and his romantic interest, and Chibiusa is the result of their devoted love for each other. Ask any parent whether they love their spouse or their child more - will anyone give you a straight yes or no answer?
Small Lady Serenity
In the future, the 30th century to be exact, Chibiusa is Usagi (Neo-Queen Serenity) and Mamoru's (King Endymion) daughter. Though we don't see a great deal of Chibiusa's relationship with her parents in Crystal Tokyo, we are given the impression that it is a good one. In Sailormoon R, we are shown a scene of Chibiusa being read a story by her father about the legendary Sailormoon. Chibiusa also dreams of becoming a beautiful lady like her mother, so it seems she has great respect and love for her parents. Though when she travels to the past, and lives with her parents in their youthful forms, her relationship is much different. Of course Chibiusa loves Usagi and Mamoru very much - the way she loves them is just different. For instance, it would seem that Chibiusa has little or no respect for Usagi - but in actuality, she has a great deal of respect and admiration for her. She just doesn't openly show it like she does with Neo-Queen Serenity.
With Mamoru, she hangs all over him, laughs with him, calls him her "boyfriend" and uses Usagi's term of endearment "Mamo-chan". I think there are two reasons for this - number one, Chibiusa views Mamoru differently than King Endymion (as explained in the above paragraphs). Number two, I am under the impression that in Crystal Tokyo, the King and Queen are sometimes forced to act a little more "proper" (even though Diana notes to Luna how the King and Queen myteriously are always sick when it's time to attend "stuffy events"), so as much as Chibiusa would love to climb all over her father, she might not be able to do so as freely. However, while living with Usagi and Mamoru, it is quite easy, so she takes advantage of that.
At the end of Sailormoon R, when Usagi has transformed into Neo-Queen Serenity to battle the Death Phantom, Chibiusa weeps in Tuxedo Kamen's arms. She knew that because she had lost the ginzuishou, Serenity had to fight. From her tears, the ginzuishou from the future appeared. We are then shown, as Small Lady joins her mother in battle, that Chibiusa is already strong enough to handle the power of the ginzuishou. This ability shows her heritage.