Jurai is a planet within the Milky Way Galaxy, as well as the capitol of the largest empire there.
Tenchi Muyo! Ryo-Ohki and Tenchi Muyo! GXP
In the official continuity, the goddess Tsunami and her sisters Washu and Tokimi, the Chousin, searched for eons for a being superior to them. Tsunami descended to the third dimension to search for this kami (God).
150,000-years ago, the giant, sleeping humanoid form of Tsunami was found on the planet of Jurai by a man who would eventually become its first emperor. After waking up, Tsunami detected evidence of potential in this man. Tsunami theorized that his descendants would produce the kami the Chousin were looking for. Thus Tsunami made a pledge to the man of House Masaki to protect him and his descendants and bless them with great power -- her descendants, the Royal Trees. The mighty power of Jurai made way for the planet and its rulers to become a galactic superpower.
Jurai (樹雷星, Juraisei) is ruled by a royal family, the Imperial House of Jurai (樹雷皇家, Juraiouke), all of whom take "Jurai" as part of their names after bonding with one of the Royal Trees, seeds of Tsunami's power which grow into sentient trees and bond to ships of the Juraian Fleet. Over time, four separate houses or families formed; Amaki, Kamiki, Tatsuki, and Masaki. The most notable of them in the series is Princess Ayeka Masaki Jurai, who is in love with the anime's main character Tenchi Masaki. She is the daughter of the Emperor of Jurai, Azusa Masaki Jurai. Azusa Jurai's two wives are the 1st Empress, Funaho Jurai, an earthling and 2nd Empress Misaki Jurai of the House Kamiki, who is also the mother of Ayeka and Princess Sasami.
The current heir to the throne of Jurai is the son of Emperor Azusa and First Empress Funaho, Prince Yosho Masaki Jurai aka Katsuhito Masaki.
Other notable members of the Jurai family are Second Empress Misaki's parents -- Utsutsumi Kamiki Jurai, head of House Kamiki and his wife, Seto Kamiki Jurai, the infamous "Devil Princess of Jurai," an influential figure in galactic politics as well as matchmaker.
Seto is also the reason House Masaki (headed by Funaho) are current rulers as she stopped the house from becoming extinct when numerous members kept running away. House Amaki is currently a bitter rival of House Masaki. The intentions of House Tatsuki are still unknown.
The planet Jurai is home to a species of intelligent trees, Royal Trees of the Imperial House (皇家の樹, Oukenoki), descended from the goddess Tsunami. In fact, the name "Jurai" is a combination of the kanji 樹 tree and 雷 thunder. These trees are capable of bestowing great powers and long life to those Juraian royals who are partnered with them, and are activated by special devices known as keys. These trees are also used to provide power to Juraian spaceships. All power supplied to the tree ships comes from Tsunami-no-ki, Tsunami's tree form. from a subspace network.
Inheritance to the throne is determined by connection to a Royal Tree; the higher the Tree is in status, the higher someone in the succession line. This is why Yosho is first-in-line, due to his connection to 1st-Generation Tree, Funaho. Tenchi's powers as a bearer of the Light Hawk Wings, which are usually generated by Royal Treeships, allow him to be second, as well as his status as Yosho's successor. This is why Seina is 3rd-in-line, as he became connected to a seed of a 1st Generation Tree. Assumingly, Sasami could surpass them all because she's connected to the original Tsunami-no-Ki, but she has not claimed inheritance on these grounds as of yet.
Juraians are also feared throughout the Galaxy for their superhuman strength, plus the ability of Juraians of noble heritage to summon forcefields and discharge energy blasts, as demonstrated in the True Tenchi Novels where a trained Galaxy Police officer was not considered a threat to even regular civilians.
Juraians are also known for their extremely long life spans, as Ayeka appears to be about the same age as Tenchi Masaki (her grandnephew), even though she is over 714 years older than him. Juraians are granted their extended life spans by their connections to the Royal Trees. In a Q&A session, series creator Masaki Kajishima confirmed that "normal" Juraians have the same lifespan as normal humans. Though Nobuyuki Masaki and others who have Juraian blood but are not, (as far as we know), bonded to a Royal Tree, have expanded life spans, though this may be attributed to their training and body enhancement while at the Galaxy Academy or his ancestry stemming from Prince Yosho.
Prince Yosho and his grandson Tenchi, are first in line for the Juraian throne though Yosho fled his throne for a life on Earth. Ayeka is still Princess but she is second in line as she and Sasami come from a distant branch of the Imperial Family and can only assume the throne if Tenchi refuses and/or she finds a husband. The members of the Imperial Family wield the mighty power of Jurai through unknown means. The members of the royal family retain their ownership of powerful treeships such as Ayeka's Ryu-Oh whose sap has the power to heal, though it is unknown if they have as many secrets to their power as in the official continuity. In this continuity, Kagato is also apart of the Juraian Royal Family but was banished to the darkest corner of the universe for being too power hungry after being defeated by Yosho. He returned and took on the missing Yosho's guise to take hold of the throne. At the end of the series, after the defeat of Kagato, Ayeka and Sasami return to their place in the Imperial Family, but leave their home to return to Tenchi on Earth. It is unknown who rules Jurai in their stead.
Tenchi in Tokyo
Besides Ayeka and Sasami being the princesses of the planet, there is not much known or said about Jurai except for flashbacks of the main antagonist, Yugi. Yugi was a Juraian child with extraordinary powers and the ability to create demons and have her will done. As an underdeveloped child, she caused chaos on the planet and the then empress Hinase was forced to seal her away on Earth, knowing completely well that she was terrified of the dark. Yugi was released sometime before the beginning of the series, and is now twisted and wants to create her own world on Earth and attempts to distract Tenchi (who has no relation to Jurai in this continuity), the latest Planet Guardian, in order to do this.
In both Pretty Sammy spinoffs of Tenchi Muyo, the name Jurai is instead used for the magical realm called Juraihelm, although the two are vastly different.