Trickster
This is your default style due to the fact that Capcom considered it the most friendly towards the newer plays to the Devil May Cry series, it's pretty much the best all round style available at the start of the game. Trickster is primarily based on defensive manoeuvres that allow you to evade and retreat from oncoming enemy attacks as you please, but after some levelling up it begins to show how truly flexible it can be by allowing you to be on the offensive from time to time. The AirTrick technique is a great example, this ability enables you to instantly teleport to an enemy's location with ease, allowing you to land a quick hit before they can even blink!
Swordmaster
The Swordmaster style is focussed on the "up close and personal" Devil May Cry fans as it forces you to use your devil arms to their maximum potential and even unlocking a few extra exclusive attacks to your moves list. You'll notice that you will be able to perform some incredibly daring, jaw-dropping combos and really rack up your style meter when used in conjunction with some of the faster weapons such as Cerberus of even Beowulf.
Gunslinger
The Gunslinger style is arguably the most flashy style that you have available, some of the moves you can pull off look truly spectacular such as the "Bullet Rain" technique and the incredible "Twosome Time" ability. Not only does this style implement extra attacks to your already ample moves list, but it also drastically increases the power of your firearms, allowing you to "charge" your favourite weapon (such as ebony and ivory or the shotgun) and release a massive blast of energy at your target, dealing alot more damage then it would normally.
Royalguard
This unique style allows your character to easily block and counter an opponents' attack with ease, as well as earn yourself a few bonus orbs and style points in the process. With the press of a single button you can repel an enemies or even a boss' attack and knock them off guard, allowing you to take advantage and dice them up while they are vulnerable.
When you successfully block an attack, you will build up what's called a "Release Attack" which is used as a gauge to determine how much damage you have absorbed and enable you to harness it and deal it out as a concentrated and potentially deadly attack upon your foe.
Quicksilver
Quicksilver is a great style for taking advantage of multiple opponents and allowing you to escape sticky situations. It basically allows Dante to move at super human speeds (although it actually just makes the enemies appear incredibly slow and you remain at the default speed) and uses your devil trigger gauge to determine how long the effects will last.
Quicksilver is potentially the most powerful style in your arsenal, especially when used in conjunction with Super Dante (unlimited devil trigger), although this could be considered "lame" amongst a few crowds. You gain this style after successfully defeating Greyon, who is also capable of using it's time manipulating capabilities.
Doppleganger
The Doppleganger isn't acquired until quite late in the game but it certainly stands out as one of the more interesting styles at your disposal. It effectively enables Dante to create a duplicate of himself (or "doppleganger" if you prefer) and allows you to control both simultaneously, dishing out twice as much damage as you normally might.
Although Doppleganger will drain your devil trigger meter while in use, it will force you to make the most of your time with it, but fortunately you will be pleased to know that you will be transformed into your devil state automatically and this will last until your devil trigger runs empty.