Spoiler warning: Plot or ending details follow.
The Titan Dweevil is the final boss in Pikmin 2. It is the largest member of the Dweevil family, and arguably one of the largest creatures in the game. But unlike other dweevils, it has no natural attack, at least that is shown; it is only really dangerous because of the weapons it carries.
At the very bottom of the last dungeon in the game, the Dream Den, Olimar and The President find Louie sitting atop a mound of treasure. When they approach the mound, however, it reveals itself as the head of the Titan Dweevil. The enormous monster then proceeds to attack you, triggering the boss fight.
Weapons
Flare Cannon
The Flare Cannon is the easiest weapon to avoid; it simply sprays flames in a set radius, as it rotates back and forth in about a 180-degree motion. When this weapon is damaged, it will spray more of the area than normal. This fire is irksome in the fact that it is capable of knocking down Olimar even if you are fireproof, stopping him from saving Pikmin.
Shock Therapist
Possibly the toughest weapon to avoid, it's the only weapon that kills on contact with the attack. When this weapon is used, it will spit out eight electrical "nodes" that will activate and form multiple arcs of electricity between them that electrocute all Pikmin that touch them. Keep your Pikmin in a tight group to avoid getting electrified. When this weapon is damaged, the nodes will activate the moment they are released. This weapon may actually reach a bit beyond the Titan Dweevil's arena on rare occasions, so it is advised to keep your Pikmin near the exit geyser.
Monster Pump
When this weapon is used, it spits out blasts of water that will get latched onto the Pikmin's heads and suffocate them. When the weapon is damaged, it will spit out water at a greater range and faster speed. This weapon's range extends beyond the circular arena where the Titan Dweevil is fought, so be sure to protect the Pikmin that are not fighting. Because of this, get rid of this one first.
Comedy Bomb
When this weapon is used, it will spray three streams of poisonous gas. The streams move at a constant speed and direction, so standing in between them at all times should be no problem. When it is damaged, it will spray four streams that may change direction spontaneously and randomly.
After the Weapons
After all weapons have been broken off (it cannot pick them back up), it will lose its armor, but still have Louie on its head. It will turn pale yellow. All you must do now is throw Pikmin at its body. It does not have any counterattacks at this stage other than shaking off Pikmin and deflowering them, but it does have a substantial amount of health. When defeated, the Titan Dweevil will seemingly melt into a bubbly, bluish goop, leaving only Louie. The weapons and Louie can be picked up as treasures. If you decide to fight this beast again, it will be without any armor or weapons (that is, if you did pick up the treasures, otherwise it will still have them). As a bonus, if you defeat this creature when it is petrified, it will drop 10 random sprays.
Notes
Weapon usage
Petrifying the Titan Dweevil while it is using a weapon results in the resetting of the time the Dweevil uses the weapon. As such, the time the creature attacks could be extended eternally, assuming infinite sprays were available.
In contrast, the Dweevil can be prevented from using its weapons alltogether, simply by quickly walking off of the arena and onto the sidelines while the creature is roaring and bobbing its head. It will not use a weapon and instead continue stomping. However, this is extremely time consuming, and only advisable in zero-death runs, if at all.
Strategy
A Titan Dweevil with all of its weapons, turning blue as it readies the Monster Pump
This boss has a rather complex way to defeating it. It has four weapons which double as treasures after it's defeated; the Flare Cannon, Shock Therapist, Monster Pump and Comedy Bomb, representing the game's four elemental hazards. It will stomp around and act like the various Arachnorb family members in their manner of stomping, except that it doesn't crush Pikmin because it has no large "feet". After it bobs its head down, it will grab two weapons, but only use one. The color it glows and which arm it raises will indicate the weapon it will use, however. The main point is to attack the weapons. If a weapon gets damaged, it will start smoking and will behave differently when used. The weapon will break off when it receives enough damage. Louie is found on top of it the first time you battle it; the Sales Pitch on him as a treasure suggests that he may somehow be in control of the Dweevil (although if you go back in the dungeon and fight the boss after rescuing Louie, it will still attack you if you didn't take the weapons last time). Despite the fact that Olimar should be immune to all hazards by the time you fight this boss, the weapons can still stun him, but do not hurt him.
Note: In case the cave was entered with a mixture of all types of Pikmin, it is advisable to still only attack with Yellows. The other Pikmin should be huddled into a corner, such as behind the geyser outside the 'bowl'. If the Monster Pump is used, the captains should be switched rapidly while blowing the whistle in order to prevent any Pikmin from drowning.
Do not underestimate the power of the Ultra-spicy spray when attacking. Since it's likely that you will only have about 20-25 Yellows, spraying them before launching an attack increases your chance to break off the weapon. You can gather lots of Ultra Spicy Spray berries in the Awakening Wood.
If the Titan Dweevil is sprayed with an Ultra Bitter Spray when it's using a weapon, the weapon will keep firing while the Dweevil is frozen. This increases the length of time the weapon's effect lasts, and the boss is much harder to damage while encased in stone, so petrifying it is not recommended until the very end, in an attempt to obtain multiple sprays.
Yellow-only strategy
The Titan Dweevil.
100 Yellow Pikmin are brought into the Dream Den. All treasures are ignored, and the player works his or her way right to the Titan Dweevil. This will help avoid loss of Pikmin from trying to get other treasures. Some sublevels have rocks over the holes that will require skill in order to get the Pikmin to these, as there are many hazards the Yellows are not immune to. The reason to use Yellows is due to the fact that they can easily reach the weapons even if the Dweevil isn't using them, and are immune to the only attack that instantly kills any Pikmin. The captains should always be kept away from the Flare Cannon. If the active one gets hit, he won't be able to escape the attack, and there'll be no saving the Pikmin.
The Monster Pump should be destroyed first. When it is used, the Pikmin should be huddled close together and continually whistled at. Attacking the Shock Therapist to the degree that it smokes but does not fall off ensures that the enemy will use the, in this case harmless, electric attacks more often, then attacking the Monster Pump. Once the Pump has fallen off, the Flare Cannon should be attacked, then the Comedy Bomb.
Keeping the inactive captain seperated from the active one makes it possible to resort to the former in case the active captain is knocked down. However, although directly under the Dweevil is the most handy location to place the inactive one, he will suffer from the blasts made by the Comedy Bomb.
Two-Time Strategy
This strategy involves going through the Dream Den and obtaining all treasures, then, when the player reaches the Titan Dweevil, only knocking off one weapon, then spraying the Titan Dweevil with Ultra-bitter spray to immobilize it and prevent it from using its weapons while the Pikmin carry the treasure back to the Research Pod. The next time the player enters the Dream Den, they do not have to worry about obtaining the treasures and can work their way to the Titan Dweevil. Each time the player fights the Titan Dweevil, they only knock off one weapon, petrify the Titan Dweevil, and exit via the geyser. This continues until the Dweevil has no weapons left and turns pale yellow. (Consult the above section "After The Weapons"). Although this strategy takes a long time, it is useful for zero-death runs.
Spoilers end here.