The second odyssey happens about 10 years after the action in the first book, the Cosmic Baby comes back, now as an emissary/servant of the mysterious god-like aliens that left the monoliths 3 million years ago. This book follow the events of the film, not the actual first book, so the action is centered on Jupiter, not Saturn. Russian and American expeditions get on a Russian spacecraft "Leonov" (after Alexei Leonov, first human to conduct a spacewalk). They want to go to the previous spaceship "Discovery" and recover anything that was left, including the disconnected HAL 9000, and also to examine the much bigger monolith (called "Big Brother" or "Zagadka") which created the "Cosmic Baby" in the previous book.
The Chinese send a spacecraft of their own, to overtake the Russians and Americans, and get to "Discovery" before them (and claim it as their own). Since they had to take less fuel to make haste, the Chinese spacecraft is supposed to refuel on Jupiter's moon "Europa", but that does not pan out when it turns out that under the ice of Europa is not only water, but living beings, not microscopic, but huge, which destroy the spaceship.
This is the main plot of the book, alien life forms on Europa, which the god-aliens of yore are trying to save (although one might wonder why it took them 10 years), by making Jupiter another sun (salaciously named "Lucifer"). The new sun thaws the ice on Europa and creates an atmosphere (not Oxygen/Nitrogen based), which accelerates the evolution on the planet, and in only 20,000 years, the "Europans" walk on the surface, have language and culture and are fully fledged sentient beings.
Overall, this book is much more boring than the first one, since the supercomputer going rogue plot is absent, and that plot is what made it huge success, and relevant even today. The plot of godlike aliens "seeding" and "weeding" planets with life is a very tired trope and somewhat of an escapism, religion-like, putting ones hopes in a higher power which should solve all of our problems including our unavoidable, final and immutable cessation of existence.
Arthur C. Clarke died in 2008 at age 90, so 26 years after this book, but it is obvious that his unavoidable death was a deep inspiration and influence on writing this novel.