Before his death, The Dark Knight Sparda fell in love with a human woman named Eva, who gave birth to identical twin sons, Vergil and Dante. Half-demon and half-human, the twins were born with supernatural abilities as well as white hair and blue eyes, all traits derived from their father's strong Demon blood. They are collectively referred to as the Sons of Sparda.
According to the Devil May Cry 3 (manga), the sons of Sparda spent their early childhood living together with their mother, Eva, after their father's disappearance. Not much is known about this part of their lives, although there are hints in the games that the twins were close as children.
On one of their early birthdays, each twin received half of the Perfect Amulet from their mother. The amulet was used by their father to seal off the Demon World and is an important key to the gateway between the two worlds. Both twins wore their half of the amulet around their neck through early adulthood, after which the whole amulet was passed to Dante following his brother's death.
Some time during the twins' childhoods, Eva was killed by demons sent by Mundus. Although the details of this event were never clearly shown, Dante appeared to have been present during the attack and witnessed his mother's death (while mistakenly believing that his twin had also been killed). In reality, Vergil was separated from his brother and mother during the attack and ended up alone in the demon world with only Yamato to aid him. This forceful separation and the trauma of being alone in the demon world may explain his cold and withdrawn nature later in life.
Eva's death was a huge cornerstone in the twins' lives and affected them deeply, leading the brothers to choose very different life paths. Dante dedicated the rest of his life to demon-hunting and protecting innocent humans, while Vergil, perhaps believing that humans (like his mother) died because they did not have the power to protect themselves, spent the rest of his life pursuing Sparda's power and trying to gain as much of it as possible.
Dante, the younger twin and the main protagonist of the entire series, always wears red leather outfits and prefers to fight with firearms. His personality is loud, sarcastic and abrasive. He often jokes around in tense situations and is so laid-back that he comes off as lazy or rude. He loves pizza (and in the anime, strawberry sundaes), which he'll consume even in the midst of battle. He jokingly flirts with most women and is shown in Devil May Cry 4 to like reading magazines with scantily-clad girls on the covers.
Dante's fighting style is flamboyant and often improvised depending on his immediate surroundings. He loves taunting his opponents and showing off fancy moves during battle. He usually uses his favorite broadsword Rebellion (an heirloom from his father, Sparda) and his self-customized handguns Ebony & Ivory, though he has also used a diverse set of weaponry ranging from an electric-guitar to a Demonic briefcase. (A running gag in the series is that he is often impaled by his own swords in battle (usually Rebellion) and brushes the wound off like it was nothing thanks to his superhuman resiliance and healing.)
Due to his closeness with his mother and her untimely death, Dante hates evil with a passion and became a Devil Hunter in order to "get revenge on the scum who murdered [his] family". He is noticeably bitter about his father's absence and his inability to prevent Eva's death. In Devil May Cry 3, he is shown to be in complete denial that he even has a father - when prompted by Vergil, he says "Father? (Laughs) I don't have a father." This attitude somewhat continues in Devil May Cry (although he admits he's Sparda's son after defeating Phantom and when confronting Mundus), although by Devil May Cry 2 and Devil May Cry 4, Dante seems to have accepted his heritage.
Vergil, the older twin and the main antagonist of Devil May Cry 3, takes after Sparda's aristocratic side and shows heavy Asian influences in the way he dresses and fights. He prefers to wear blue outfits made of silk and slicks back his hair into loose spikes, much like his father. His personality is cold, calculating, ruthless and withdrawn, although he occasionally shows a dry sense of humor when annoyed. He views guns as unworthy of a true warrior, and so only uses melee weapons like his heirloom katana, Yamato, or the Beowulf Devil Arm. However, he also uses magically-summoned phantom blades which bear little functional distinction from firearms.
Vergil's fighting style is notable for its speed and range. He is capable of teleportation (although Dante is a better jumper) and can move his sword faster than the eye can see with his superhuman speed. For most of Devil May Cry 3 and the Devil May Cry 3 (manga), he is shown to be a stronger fighter than Dante. Even after his defeat at the hands of Dante and Mundus, he returns as Nelo Angelo in Devil May Cry and is still capable of disarming Dante in battle. Interestingly, he is an honorable warrior - aside from his dislike of guns in general, he despises fighting dirty and (as Nelo Angelo) refuses to attack opponents from behind.
Whereas Dante is focused on protecting Humans from Demons, Vergil is obsessed with obtaining power. He greatly admires Sparda and wishes to surpass his father in strength and righteousness. In Devil May Cry 3, he was even willing to undo his father's work and risk Demonic invasion in order to become stronger. Since he believes that Humans are weak, he possesses an extreme dislike of his Human side and fully embraces his Demon side, often referring to himself as a Demon and calling on Dante to do the same. In Devil May Cry 3 (manga), Vergil forcefully unleashed Dante's Devil Trigger for the first time in an apparent attempt to force his twin to embrace their Demon heritage.
A major driving force in the Devil May Cry storyline is the duality between Vergil and Dante and how different they are despite being identical twins. They are polar opposites in tastes, fighting styles, personalities and goals and spend much of Devil May Cry 3 (manga), Devil May Cry 3 and Devil May Cry fighting with each other. They seem to represent the two sides of Sparda himself - he was known to be both rebellious like Dante and level-headed like Vergil, and his preferred color, purple, is the combination of Vergil's (blue) and Dante's (red).
Although the brothers are always at odds, it is heavily implied that the twins were once close in their youth and (to a certain extent) still care for each other. During the battle against Arkham near the end of Devil May Cry 3, the twins fight as a team and are comfortable pulling off complicated maneuvers like swapping swords in mid-air. They even share Ebony and Ivory at one point and shout "Jackpot" as they shoot, which Dante claims is something they used to say as children. This scene is the first and only time Vergil is shown to use a firearm. Vergil even admits this, commenting "I'll try it your way for once."
When Vergil throws himself into the depths of Hell at the conclusion of Devil May Cry 3, Dante sheds some tears for the loss of his brother in front of Lady, who later notes that there are good demons in the world capable of crying for those they care about. Dante then opens his shop and names it 'Devil May Cry,' which is a twist on the supposed fact that Devils don't cry. This implies that the name of the series is partially based off Dante's angst over Vergil's sacrifice.
Years later in Devil May Cry, when a corrupted Vergil (now the Demonic Angel warrior Nelo Angelo) fights with his brother, he is tormented by memories of his family when he sees Dante's half of the Perfect Amulet around Dante's neck. Dante realizes he is Vergil only after he defeats him, and loses his brother for the third time.
In Devil May Cry 4, Dante spends much of his time searching for something that is "very important to him", which ends up being Yamato, his brother's katana. Although at the end of the game, he allows Nero (Vergil's son) to keep it.