Mission 1: Some time prior to the Naboo Crisis, Qui-Gon aids Chewbacca and his father as they help settle Alaris Prime (a moon in the Kashyyyk system) while fending off Trade Federation battle droids.

Mission 2: Darth Maul arrives on Naboo and helps rout the Gungans.

Mission 3.1—3.3 deals with the "War of the Gungan Tribes" three thousand years in the past.

Mission 3.4—3.7 deals with getting Boss Nass and later Queen Amidala to the Sacred Place during the Naboo Crisis. Later portions deal with Boss Nass' battles during the Battle of Naboo, engagements known as "The Battle of Spinnaker" and "The Liberation of Harte Secur."

 

The storyline in the Clone Campaigns expansion is referred to as "The Hunt for the Decimator":

From the Official Site: "The fighting spreads on worlds of the Outer Rim, as the Confederacy steals a Republic prototype weapon, the Decimator, and engages the Republic forces on Eredeen, Aereen, Sarapin, Kaer, and the Wookiee moon of Alaris Prime."

 

Major engagements from this Separatist  portion (Sev'rance Tann) of the campaign include:

Mission 7.1: The Battle of Geonosis

Mission 7.2: The Kaer Orbital Platform (near Tatooine)

Mission 7.3: The First Battle of Tatooine

Mission 7.4: The Battle of Eredenn

Mission 7.5: The First Battle of Alaris Prime

Mission 7.6: Asteroid 426 (near Sarapin)

Mission 7.7: The First Battle of Sarapin

 

Major engagements from this Republic portion (Echuu Shen-Jon) of the campaign include:

Mission 8.1: The Battle of Geonosis

Mission 8.2: The Second Battle of Sarapin

Mission 8.3: The Second Battle of Tatooine

Mission 8.4: The Battle of Aereen

Mission 8.5—8.6: The Battle of Krant

 

Following these events, the two Jedi are sent by the Jedi Council to Raxus Prime, the current lair of Count Dooku, to uncover a secret project occurring there.  Upon arrival, Anakin is soon captured by Count Dooku who willingly shows him the project: an ancient Sith super-weapon, called the Force Harvester, or the Dark Reaper, an instrument capable of draining the Force from any living source and utilizing it for destruction.  Anakin is imprisoned onboard Dooku's ship with the female partner of Cyan Pratt, a mercenary who aided Dooku in obtaining the Dark Reaper from the depths of Raxus Prime.  Anakin and the mercenary manage to escape while Dooku begins testing his new weapon on the moon of Kashyyyk. 

After aiding the Wookiees in their defense (to little avail as the Dark Reaper causes the forests to be razed), Anakin returns to Coruscant to inform the council what he has learned.  The Dark Reaper, meanwhile, causes some destruction on Mon Calamari, Agumar and Bakura.  Anakin is sent back to Rhen Var to learn the secrets of the Force Harvester from the hologram (or spirit?) of the ancient Jedi Uliq Quel Droma, who had defeated it millennia earlier.  Uliq imparts secret knowledge to Anakin, warning him that such power could lead to the Dark Side. 

Anakin returns, and together with a Jedi invasion force, heads to the moon of Thule where the Dark Reaper is currently stationed.  With the knowledge he'd gained from Uliq, Anakin defeats the Sith weapon while the Jedi attack the Separatist forces.  Cyan Pratt is likewise defeated by Obi Wan.  After the battle is over, Obi Wan mentions to Anakin that he has grown more powerful.

Anakin is sent by Master Windu and Yoda to investigate rumors of the emergence of deadly new droids.  The informant, Raala Ponchar, however, is now missing.  Anakin heads to Tatooine for information and discovers that womp rats have become wildly rampant.  A mysterious figure offers him information if he'll take care of the womp rats.  Anakin agrees and discovers that anooba creatures have taken over the womp rats' caves.  He eliminates the deadly creatures and heads back into town.  The mysterious figures turns out to be the Jedi killer Aurra Sing who wanted Anakin dead.  Word spreads fast of Anakin's success with the anooba.  When Watto discovers Anakin is on Tatooine, he approaches him with information on Raala Ponchar.  Likewise, a townsperson informs him that Hutt assassins are in town "looking for something or someone."  Watto gives Anakin the directions to find Ponchar and jokes that it would be safer for Anakin if he took back his job with him. 

Outside town, Anakin meets a stranger who asks him for help dealing with the Tuskens who've captured several moisture farmers.  Anakin agrees to help him rescue his friends.  Anakin frees all the farmers, and the last one tells him that Ponchar worked for a farmer named Bellek.  Anakin goes to find Bellek, but discovers only a burnt down farm.  A resident of those parts tells him that a droid like no other, led by a "witch," razed the farm.  Before Anakin's eyes, the resident is killed, shot down from behind.  Anakin sees the new droids first hand;  green, super-battle droids with hand-held combat weapons and cortosis armor, which is lightsaber-impenetrable.  He manages to defeat them, however, and find Bellek who tells him that Ponchar was likely taken to Jabba's Palace.  He discourages him from going there as the shortest path is through Tusken country.  Nevertheless, Anakin tells him he knows how to deal with Tusken Raiders.  When asked about the "witch" Bellek cannot answer. 

Anakin fights his way through the Jundland Wastes and the Sandpeople to Jabba's Palace.  There, he finds Aurra Sing and confronts her.  But a mysterious woman with two red lightsabers takes Anakin by surprise with the Force, and tells Aurra to put him in the dungeon.  Awaking in his cell later on, Anakin finds Raala Ponchar.  With the Force, Anakin opens the cell door.  On their way out, Ponchar tells Anakin that Jabba was hired by Wat Tambor of the Techno Union to facilitate smuggling operations of construction materials and cortosis.  Shipments were headed to Metalorn.  Ponchar and Anakin go their separate ways and Anakin takes some time to free other farmers on his way out of the dungeon.  But he is pursued by the "witch" with the two lightsabers.  She is Dooku's Dathomiri apprentice Saato.  In battle by the Sarlacc Pit, Anakin defeats her.  Anakin transmits a message to Obi-Wan with information.  Obi-Wan requests his immediate return because Coruscant is under attack. 

Anakin arrives on Coruscant shortly and goes into battle against the new droid army with his fellow Jedi.  As Obi-Wan departs to fetch reinforcements, Anakin senses a disturbance in the Force.  He is approached by a mysterious figure who identifies himself as Trenox.  Trenox tells Anakin he should rethink his path, as the Jedi are at an end.  Anakin pursues the fleeing Trenox to the lower levels of Coruscant and there discovers Bariss Offee.  Offee's clone troopers were overrun by Separatist Droids.  Anakin next bumps into Dexter Jettster who says that en route to pick up supplies, he was threatened by a Jedi with a red lightsaber.  Anakin goes to check the docking bay for Trenox, but discovers instead a blue alien in a purple outfit.  He tells Anakin that Trenox has probably already destroyed the Temple.  Anakin races back to the Temple only to discover Count Dooku who comments on Anakin's new arm.  Dooku declines a battle with him, saying he is going to destroy the Jedi Archives.  He taunts Anakin that he would likely the choose the same path as he to end the war, but Anakin exclaims "I will never be like you!"  Obi-Wan contacts Anakin and informs him that the battle is a ruse, and that he'll soon arrive with reinforcements.  Anakin departs to the Jedi Archives and finds Trenox there.  He fights and kills him.  Yoda then contacts Anakin to inform him that reinforcements have arrived, but Dooku has escaped. 

In a meeting with the Council, Anakin discovers that Obi-Wan was hurt in the battle.  The Council tells Anakin to go to Metalorn to retrieve the schematics for the cortosis battle droids.  He is to arrest Wat Tambor and bring him back to Coruscant unharmed.  Anakin soon infiltrates the factories on Metalorn and contacts Yoda.  Yoda instructs him to destroy the new droid army there as an army of cortosis droids would be a great threat to the Jedi.  In the depths of the factory, Anakin is confronted by two Dark Jedi, the brothers Vinoc and Karoc, but Anakin defeats them.  With detonators found in the factory, Anakin plants them on the reactors and confronts Wat Tambor.  Tambor refuses to go with him and introduces his bodyguard, Vandalor, a Chiss darksider who was the blue alien Anakin had met earlier.  Anakin manages to defeat him as well and arrests Tambor who mourns the destruction of his new droids.  Before Anakin can depart, however, he senses a disturbance in the Force.  It is Count Dooku.  As they fight, Dooku taunts him and tries to get him to unleash his anger by bringing up his mother's death.  Anakin defeats him, but Dooku claims triumph for giving him a taste of the Dark Side.  Dooku disappears (it is later revealed he was not the real Count Dooku, but a doppelganger devised by the dark side of the Force).  Back at Coruscant, Anakin informs the Jedi Council that Wat Tambor was taken into custody by Palpatine's personal guards, but Mace finds this curious (Wat Tambor later manages to escape).  The Council is impressed with Anakin's growing powers.  Yoda also warns about the future, "Vigilant we must be now.  A growing shadow of darkness I sense.  Weary from war the Republic grows."  Anakin vows to always be there to serve the Jedi.

Note, at least one element of this game has appeared in a published novel (Yoda: Dark Rendezvous) referring to Anakin's fight with Count Dooku (the Dooku clone/doppelganger) so the events of this game are considered historical from a chronological perspective.

Another excellent synopsis of the storyline of this game appears here!