Peaches, like apples and pears and avocados and tomatoes among other fruits, can ripen after being picked. This comes in handy both because if you accidentally end up with unripened peaches, it's nice to know that they can be saved, and because you can purposefully choose peaches at different levels of ripeness with the plan of ripening the harder ones to eat later. There are three ways, each with its own speed, to ripen peaches once you get them home.
A quick note: Peaches that are picked too green will not ripen, no matter what you do. To see how to choose peaches that will ripen, see How to Buy Peaches.
First, the easiest method is simply to leave peaches on the counter. Peaches like best to be kept on their shoulders (not their bottoms) and not touching one another. That may not work for your kitchen, and putting them in a bowl isn't the worst thing in the world, just handle peaches gently and don't stack too many on top of others. To speed this up, you can leave them where they will receive some direct sunlight, as long as that spot doesn't get to hot and start baking them.
Second, if you want to speed things up, pop the peaches in a paper bag. It will contain the ethylene gas the fruit gives off, which hastens the ripening process. Even faster? Add a banana in there, too, for more ethylene action.
Third, when you want to vastly slow down ripening, put the peaches in the fridge. Cold slams the brakes on the ripening process, but doesn't bring it to a complete screeching halt. Check on peaches in the fridge daily; if left in there too long they can start to dry out.