Certain style feats can be used with weapons, such as Monkey style (the first level of it). Most styles require you to fight unarmed or even give you natural attacks to fight with. Some styles actually give you flat bonuses and bonuses to your unarmed making them still useful with weapons.
Edit; here is a list of the styles and a bit of there uses;
Boar: increased unarmed damage, bleed, good intimidate (unarmed only, Improved Intimidate)
Crane: good defensive fighting, increased deflection if you have at least one hand free (any, mostly 1 hand free at least)
Djinni: Increased elemental fist usage and power (unarmed only)
Dragon: First gives you bonus to saves, mobility unarmed damage. second/third gives extra unarmed damage, breath weapon (1st level any weapon, 2nd/3rd unarmed only)
Earth Child: Bonus dodge and trip against giants, Stunning Fist against prone giants (any weapon, must be dwarf or gnome, small bonus to unarmed)
Efreeti: Increased elemental fist usage and power, fire resist, and cone of fire (unarmed only + cone of fire)
Janni: First gives you bonus to charge, and being flanked, second/third improved bullrush/trip and charging (1st level, any weapon 2nd/3rd unarmed only)
Will add the rest later, as I can. Right now Im going to bed, and Monkey style is my favorite. You can function normally while prone and works great for the goblin feat Roll with it.
It depends on the Style feats in question. The Boar Style tree explicitly calls out using unarmed strikes, while the Crane Style tree only requires 1 hand to be free. So one could use Crane Style while wielding a longsword, and no shield, but not when wielding a greatsword, or two-weapon fighting.
As others have said, depends on the Style in question. Dragon Style is a great pick-up feat if you already have Improved Unarmed Strike for some reason as you can adopt the stance as a swift action and ignore difficult terrain for your Charge attacks.
You will have to read the text for each individual Style to determine the applicability with weapons. Generally speaking, the attacks themselves are Unarmed Strikes, but secondary effects (like the ignoring of difficult terrain on Dragon Style) are fair game for everything.