"Problem" may not be the right word
to describe this, but device variance is an issue you need to consider when developing an Android application. Android runs on a wide variety of device hardware and along with that comes a handful of different screen size/resolution/aspect ratio combinations. I would say that the number of device screens to support does not even begin to match the number of Android devices on the market, most all devices (certainly those running Google Play) will fit into about 2-3 types for handsets and 2-3 types for tablets.
The native Android framework works very hard to provide you with the tools to develop your application's user interface in a way that will be flexible and scale to accomodate the different screens. The more you can utilize the resource framework Android provides to select appropriate layouts, images, dimensions, etc. that best fit different screen types, the less work supporting the Android ecosystem will become for you as a developer. This means, above all else, designing your UI in a flexible manner. Here are a few thoughts on this:
You can fix the size or position of an element, but not both.
It is okay to provide fixed assets in your UI (i.e. a button must be this size to fit the background I've created for it, or this element should always be 10dp from the top of the screen) but avoid trying to do both. If you must fix the size, allow the location to float appropriately so that extra space on larger screens is used appropriately.
Where possible, use scalable image assets
Creating a static image for a button background may not be the best approach if you want that button style to wrap the text you put in, as that background will stretch and skew in many cases. The Drawable class and 9-Patch graphics are your friend here that can allow you to create graphics in your application that are not dependent on a fixed pixel size.
Use scaled dimensions always
In the first point I mentioned "10dp", which is a scaled dimension unit
Android provides. By using these values to declare fixed sizes/positions instead of direct pixel values, Android will do the work of making that dimension look correct based on the density of the device's screen.
In response to your last question, there are several recipes in Chapter 2 that assist you in seeing how the resource system and flexible graphic assets can help you build a great native application that isn't dependent on any one device type.