Warning: Spoilers follow.
For both the book and the movie, the story is set nine years after the Discovery mission to Jupiter failed. Note that the original 2001: A Space Odyssey featured the journey to Saturn instead.
A joint Soviet-American crew, including Heywood Floyd from 2001, on the Soviet spaceship Cosmonaut Alexei Leonov arrives to essentially discover what went wrong with the disappearance of Bowman into the "stargate", or TMA-2, and to unravel the mysteries behind the monolith in orbit around the planet on the satellite Iapetus. This information is locked away on the now abandoned Discovery craft - the Soviets have an advanced new drive, the Sakharov drive which will propel them to Jupiter first, so Floyd is assigned to the Leonov crew as part of a joint mission. However, a Chinese ship, the Tsien (originally thought by others to be a space station built in orbit), is racing at a headlong speed towards Jupiter also. The Leonov crew on their mission comment on the kamikaze-like method of the Chinese team, but Floyd eventually surmises that due to the large water content of Europa, they are destined to land there and use the water content to refill their tanks.
The Leonov eventually dock with the Discovery, and Hal's creator, Dr. Chandra, on the mission, restarts the HAL 9000 computer to ascertain any information HAL has.
An apparition of Bowman, whose spirit (the Stargate has transformed Bowman into a form similar to that of the creators of the Monolith, a "star child"), forms in the dust, in the novel, warning Floyd that they must leave Jupiter within seven days. Floyd has difficulty convincing the rest of the crew, at first, but a dark spot on Jupiter begins to form and starts growing.
Unfortunately they must leave HAL behind on Discovery. The crew are worried that the computer will have the same neuroses on discovering that HAL will be abandoned yet again, and install a hardware failsafe, secreted from Dr. Chandra (however, which he discovers and disconnects). All is well, in the end, but Bowman speaks with HAL, giving him a special mission.
The Leonov crew make a hasty exit from Jupiter, using the same aerobraking maneuvre demonstrated by the Tsien in its headlong dash to Europa, just in time to witness the engulfment of Jupiter, and the primitive life described living in its clouds, by thousands upon thousands of Monoliths, which convert the planet into a star (Jupiter is often described as a "failed star").
As the Leonov leaves Jupiter, HAL begins broadcasting the message "ALL THESE WORLDS ARE YOURS EXCEPT EUROPA. ATTEMPT NO LANDINGS THERE" repeatedly until the new star, which Earth has dubbed Lucifer (bringer of light) destroys Discovery entirely.
The sequel 2061: Odyssey Three handles the consequences of the warning and the subsequent breaking of the warning to save the lives of a hijacked crew.
Considering the impact of 2001: A Space Odyssey, this sequel is considered by many to be a disappointment. However, as a movie on its own, it is also considered to be quite good. The novel and movie do have some differences, most importantly the absence of the Tsien in the movie.