If you have one formal and one casual dining space, you can really go to town with different chair designs for both. A casual spot will do best with sociable, comfy seating that naturally leads to conversation, whereas a formal setting is where you can play around with luxury fabrics. If you only have one dining room, you're going to need to combine all of these elements in one set of chairs, if possible!
One of the first things you will have to ask yourself is where you will be placing your dining table. A round table curiously seems to draw all elements that surround it towards it, thereby anchoring itself (and the chairs) visually more than a rectangular table would. It also takes up less space.