
Background
Fuku first appeared in the episode "Neju Who?" She is a "cabbit," a small, part rabbit, part cat-like creature that hatched from an egg in the Kamidake II, the ship of GXP main protagonist, Seina and was given her name by Seina himself for the Japanese name for good luck, the complete opposite of Seina himself who is notorious around the galaxy for his bad luck. She is the younger sister of Ryo-Ohki, the extremely powerful cabbit/spacecraft of space pirate Ryoko created by Washu. Fuku has a great love for her creator/mother, Washu and has fond memories of her even before hatching.
Washu created Fuku on the request of Tenchi, who was worried about the safety of his extremely unlucky friend Seina in the Galaxy Police.
One thing to note, is that the cabbit appearance of Ryo-Ohki resulted from Ryoko integrating the remains of Chibi, a little cat she had been keeping alive for Tenchi before he released her at the beginning of OVA 1. So while Ryo-Ohki's appearance is a side-effect of the integration, Fuku's was modeled after the integrated Ryo-Ohki.

The Kamidake II

Fuku with her big sister in her humanoid form.
Fuku, as well as the Kamidake II, were also briefly taken away by a corrupt faction of the Galaxy Army that were aiding Shank in order to analyze her for their own personal desires and placed her into a stasis field. The G.A. scientists attempted to clone Fuku, but could only produce five imperfect clones. Eventually Seina was able to rescue Fuku, as well as the clones. As for what happened to the clones are unknown.
Towards the end of the series, Seina's crew and the girls who vy for his affection, Amane, Ryoko, Neju, and Kiriko were later engaged to wed Seina in a political marriage. And as the Kamidake II now falls completely under Seina's control, with Fuku as a part of the ship, Seina in a way was marrying Fuku as well.
Fuku also made a brief appearance in Tenchi Muyo! Ryo-Ohki,, when at the end of the final episode of OVA 3, she travels back to that time period from an unspecified point in the future, alters the message to Mihoshi Kuramitsu's family (that antagonist Z had originally altered in the original timeline) in order to put the timeline in correct order, so that she would eventually meet Seina. Fuku was accompanied by a slightly older version of Noike, who also didn't want the timeline adversely affected so that she would be able to meet Tenchi.
Fuku's relationship with Seina is very similar to Ryo-Ohki's relationship with Tenchi . Fuku is deeply devoted to Seina, and that devotion is clearly demonstrated by her willingness to help him, despite the fact that sometimes she can be easily frightened. Her personality is also very similar to Ryo-Ohki in the sense that she is very sensitive and desires to be loved and showed affection by others; and when she is sad and lonely, she easily becomes depressed. Seina was also the first one to truly care for her when he demonstrated his faith in her.

Fuku with Neju
As a result of her relationship with Seina, Fuku has demonstrated to be very powerful whenever Seina or anyone else is in danger, which has lead to her mood and feelings supplying immense and powerful energy to the Kamidake II whenever its crew were in danger.
Fuku is not nearly as powerful as Ryo-Ohki, as the elder cabbit's power stems from
Adult Fuku

Ryo-Ohki and Fuku in humanoid form.
Additionally, Fuku, along with Noike, appeared in the final scene of Episode 6 of the third Tenchi Muyo! OVA, making sure the letter to Mihoshi's family (originally altered by Z) got through, with Fuku doing the typing.
It is reveled to be the reincarnation of Yayoi Schwael in the Paradise Wars novels. Also, in the GXP Light Novels, it is revealed that Fuku has the soul of a mass murderer named Widow, whose soul was captured immediately after her execution, and used to help create Fuku.