Wider tires add comfort, but aren't necessarily more puncture-proof than narrower tires. Keep in mind that you need to buy a tire that will fit your bike. My road bikes, for example, can't handle anything wider than 700x28 because the tire won't squeeze past the brake pads.
If you're only going to ride on paved roads, look for a slick tire. I've had good luck with Continental Ultra Gatorskins. Didn't have good luck with Vittoria Randonneur Hyper tires, but the Pro version is supposed to have better flat protection.