It is (as of Chapter Two) the year, cycle, and Great Cycle of the Spider, which means the Spider can act directly on each Day and Month of the Spider. On the early Day of the Spider, the PCs were flung throughout the lands. On the Day of the Grasping Spider, we believe the Spider acted to kidnap Li Kao. We do not know what the Spider did on the late Day of the Spider. Spider will be able to act directly three more times, at the end of the year.
Yoshi used his Sense Motive schtick on the manifestation of the Spider in Monarch Butterfly and learned that Spider wants Power and Dominion. This quest for power is probably sourced in the fact that Aku is a demon, and it trickles down to many humans who are touched by the Spider. Normally, one would expect those touched by a "regular" Spider spirit might be less focused on sneakiness and power-grabbing, and more focused on, say, creation.
The Spider replaced the Dragon in the Cycle at the beginning of this Great Cycle of the Dragon/Spider, 156 years ago. Spider may have been involved in the death of Lisan Tan, the last Emperor, and was almost certainly involved in the creation of the Regency Council. Since then, he has enacted plans to rename the Twelve Kingdoms, redirect the chi flows of the Empire into the World Above (possibly into himself), and, we suspect, create the Empire of the Spider. If the Spider remains one of the Cycle Spirits through the end of this Cycle (i.e., the next twelve years), the change will be permanent.
Spider has made bargains with at least some of the other Great Spirits; Phoenix will not act directly against Spider, and Spider will not kill the Dragon. If the Dragon is freed (by the party, say), then Phoenix will be free to act.