Pet: Indoors only. Not a pet (e.g., barn cat, mouser, whatever -- you don't have any attachment to them): Outdoors okay.
Indoor-only cats live a lot longer than indoor/outdoor cats -- on average twice as long. There are a lot of things that can end an outdoor cat's life, such as getting hit by a car, predators such as foxes and coyotes, poison (rat poison, antifreeze, etc.), dogs, cat fights, etc. Indoor cats, on the other hand, are much less likely to be poisoned, won't be hit by cars or killed by predators, and will most likely live out long healthy lives until old age or disease takes them.
The argument I usually get from people is that it's more natural for cats to let them outside. I call B.S. on this. First of all, nothing about domestication is natural. It's not any more natural to eat food from a can or a bag, poop in a litter box full of chemicals, or get spayed or neutered, yet most cat owners do all three of those things without a second thought. It might be more natural to let your cat starve to death if they can't figure out how to catch a mouse, but that doesn't mean you're going to do it.
Cats don't understand cars and poison, and most house cats have no idea about predators, either. Letting your cat outside is like letting your 2-year-old play in the street. They may be fine, but then again, they may not. Personally, since I love my cats, I don't want them to leave me sooner than they need to... And if you have kids, do you really want to set yourself up for having to explain to them why their kitty isn't around anymore?