Play Store not downloading app without wifi
There are two things to check.

Go into Settings from the Play Store app's menu. At the time the question was written, the third one down is Update over Wi-Fi only. Turn this off if you want to download apps over a cellular internet connection.
In more recent Play Store versions, it's at the top of the list: it says Auto-update apps, and the default is Auto-update apps over Wi-Fi only. You need to change this to Auto-update apps at any time. Data charges may apply.

Apps larger than a certain size (I think 50 MB) only download over Wi-Fi regardless of this setting. (The Play Store app will prompt you if you try to download a large app and you're not on Wi-Fi.) This applies to lots of games but not many applications.


The reasoning behind all of these restrictions is because many users pay per kb of data, or they pay extra if they go over a limit. The default behaviour is safe and avoids surprising users with large bills.