Obligatory post describing Dante's special abilities and fighting skills. Warnings for tl;dr of course. :D
Physical Abilities
Dante is a half-demon, which means he gets nifty abilities like being able to heal nearly instantaneously from almost any wound. Shot in the head? It's going to knock him back, but it's only a minor inconvenience. Sword slash to the chest? He'll stagger, but he'll be right back at you in a second or two. (That doesn't mean he's indestructible though. If he is wounded multiple times in a short time period, it's going to weaken him and slow him down.)
He's incredibly strong and ridiculously fast. The combination lets him do things like see a bullet heading towards him and cut it in half in mid-air. He can also do some pretty neat moves (see below).
The other fun trick Dante gains from his demon heritage is the ability to use a 'Devil Trigger'. When he does this, he temporarily changes into a
demonic form, and his strength, speed, and recuperative abilities increase even more than usual. However, he can only sustain this for brief periods of time, and he can't use it if he's already in a weakened state.
Moves
His speed gives him the ability to dodge most normal attacks, but Dante also has two special moves he can use if he jumps into the air. 'Sky Star' lets him move forward while in mid-air, and 'Air Trick' lets him disappear mid-leap and reappear right over an enemy.
Dante's speed also means he can run up walls, and even when he jumps, he can use magical energy to create a temporary platform under himself. This lets him push off again and jump even higher.
The Devil Arms
Dante's main melee weapon is a huge sword named Rebellion. It's of demonic origin, and it was left to him by his father, the demon knight Sparda. He carries it
on his back (yes even when he's completely shirtless - it's apparently a magic velcro sword).
Most of Dante's in-game sword moves with Rebellion are just really powerful sword strikes. And by 'really powerful' I mean he can do an upward strike that will fling an enemy up into the air, and an overhead strike that is called 'Helm Breaker' for a reason.
There are two special moves though that are probably worth mentioning. 'Million Stab' is just like it sounds. Dante stabs an enemy multiple times "with blinding speed creating enough momentum to blast them backward". The other move, 'Drive' is "a lightning quick sword swing that sends a powerful shockwave towards enemies".
Cerberus is a triple nunchaku weapon. It can stretch to incredible lengths, and is infused with the power of ice.
Cerberus was the three-headed Gatekeeper of the entrance to the Temen-ni-gru, chained up and sealed away by Sparda. As a weapon, Dante can use Cerberus to create ice pillars and small glaciers to attack his enemies, and he can even use the ice as armor against attacks.
Agni and Rudra are a pair of living blade Devil Arms with serrated edges. They can be combined, and are able to hurl forth fire and tornado-like gusts.
Agni and Rudra were twin demon gatekeepers of Temen-ni-gru sealed away by Sparda. Agni had the power to control fire, while Rudra had the power to control wind. As weapons they give Dante the ability to create a vortex of wind or flame, or send fire racing along the ground towards his enemies.
(Agni & Rudra are also the only Devil Arms that have been heard speaking after they were transformed into weapons.)
Nevan is a scythe weapon, able to control electricity and bats. (It's also a playable guitar.)
Nevan the Lightning Witch was another guardian of Temen-ni-gru. She was a succubus/vampire demon who seduced men down the path to hell. As a weapon, Nevan gives Dante the ability to attack enemies with lightning and hordes of bloodthirsty bats. If Dante is using Nevan in his Devil Trigger form, he will sprout wings and be able to fly for short periods.
The Guns
Dante carries two custom guns, Ebony & Ivory. He keeps these holstered on his back also. They've been specially modified to allow for rapid-shot capability, so he can unleash a hail of bullets on his enemies. (And they magically reload themselves, of course, since he never seems to run out of ammo.)
Dante is really good with these guns. He can fire simultaneously in two different directions with the same accuracy. He can jump into the air and slow his fall by shooting an enemy below him. If he has time to focus, he can also charge a shot with magical energy, making it even more destructive.
Dante sometimes carries a Shotgun. This shotgun is a close-range weapon that fires buckshot and has been customized for combat against demonic foes. Dante can also charge his shotgun blasts with magical energy.
Other Tricks
Dante has two other abilities that don't require a particular weapon.
Doppelganger: "Initiate a dimensional light-rift causing a second shadow version of yourself to appear." The doppelganger basically mimics whatever Dante does, so if he's fighting, there are suddenly two Dantes fighting. They'll make the same hits at the same time (so it's like synchronized fighting or something). He can only do this for short periods of time though.
Quicksilver: "Send out a dimensional time-rift wave that slows down your enemies while giving you ultra-high speed." This only affects a small area, so it can be dodged if his opponent is fast enough. And again, he can only do this for a short period of time.