The Logrus is an underground cavern located at the exact opposite end of the universe from Amber, by definition. It appears to move around, but actually, it's moving everything ELSE around. As it wanders, it is usually to be found in House Gainax. Its configuration constantly changes.
One must possess shapeshifting to the point of being able to assume primal form in order to sucessfully walk the Logrus from beginning to end. Those who do so, gain the ability to call upon the Logrus to perform various tasks, gaining power over Shadow.
While the TAMD/UNS Pattern has yet to show any direct sentience of its own, the Logrus has been known to manifest an ever-changing squiggle known as the Sign of the Logrus, though which it communicates.
Because it represents change, the Sign of the Logrus is not the most stable of beings. Because the TAMD/UNS Logrus is in some way a manfestation of a greater Logrus, it isn't even always aware of exactly what universe it is in or who it is talking to.
It seems to especially dislike Gabrielle, for complex and poorly explained reasons (their famous Steel Cage Macth nothwithstanding) and for a time seems to have been plotting the destruction of Ember by whatever means necessary, so as to wipe Gabrielle out of existence. After a co-operative venture with Gabrielle, however, the Logrus seems to have renounced this goal.
During the Herald Crisis, the Logrus gave birth to a daughter, Jewel, and for a time turned very 50s sitcom wifey in response to this. This led to some conflict with Jacqui over the issue of Jacqui's current romantic relationships, but was apparently resolved when Jacqueline fragmented like a dropped snowglobe, sending Jack? off to live with the Logrus and Jewel.
The Logrus was anointed Archbishop of Canterbury in a strange ceremony attended by Setsuna, among others.
Mitsune has identified seven major forms of insanity commonly found among those who have just Logrus Walked:
The Sign of the Logrus manifests itself. "GReeTInGs, EArTh CReaTURes."
Justin says, "Please, umm...state your name."
"I aM thE SIGn oF tHe LOGruS."
"Were you present at the events of [Some chaos date]?"
"WHEreEVer LOGrUs USeRs go, I AM."
"Did you witness the aforementioned death of ten knights of Hendrake?"
"DAMn StRaIGht. It'S ALL GaBRIeLLE's FAult."
Justin sweats. Justin says, "In what manner?"
"PUNk ASs PATtern. HeR and THAT biTCh, GidEON. TelLINg ME, whAt to dO!"
Motoko stares at the sign of the Logrus, and winces.
Shizumaru begins to sweat, as do many other Amberites present.
"Are you saying they sent the demon?" Justin asks.
"ANd THey TOok mY dAmN sanDWIch, too!"