If I develop an app for iPhone or iPad, how much does Apple charge me? What fees does Apple charge iPhone/iPad developers? Is there an entrance fee so that the app is listed in the App Store, and some charges on each sold app?
If you want to develop HTML5 web apps that run in Safari on your Mac/iPhone/iPad, it's free.  These apps can be designed as web clippings for your device as well.

If you want to develop Simulated iOS apps that run on the iPhone/iPad Simulator on a Mac, it's free (after the cost of your Intel Mac with Snow Leopard or Lion), or maybe $5 to download the tools from the Mac App store.

If you want to develop, install and run a native iOS app on any stock OS iPhone or iPad, you need to pay $99/annum to enroll in Apple's iOS Developer program.  

If you want to submit a native app or an app encapsulated HTML5 web app to Apple's iTunes App store, you need to be enrolled in the same $99 per year iOS Developer program.

If your app is a paid app that they approve for distribution from their App store, Apple pays you around 70% of what it charges the customers (less if there's a VAT tax).

Added: Starting with Xcode 7, and with a valid AppleID, you can use a personal profile to install apps developed using Xcode on your directly connected personal iOS devices, without paying $99 for an Apple Developer enrollment.  But apps installed this way are reported to have a short expiration date, around 1 week or so.