How do I get the APK of an installed app without root access?

Accessing /data/app is possible without root permission; the permissions on that directory are rwxrwx--x. Execute permission on a directory means you can access it, however lack of read permission means you cannot obtain a listing of its contents -- so in order to access it you must know the name of the file that you will be accessing. Android's package manager will tell you the name of the stored apk for a given package.

To do this from the command line, use adb shell pm list packages to get the list of installed packages and find the desired package.

With the package name, we can get the actual file name and location of the APK using adb shell pm path your-package-name.

And knowing the full directory, we can finally pull the adb using adb pull full/directory/of/the.apk

Credit to @tarn for pointing out that under Lollipop, the apk path will be /data/app/your-package-name-1/base.apk

Android appends a sequence number to the package name to produce the final APK file name (it's possible that this varies with the version of Android OS). The following sequence of commands works on a non-rooted device:

1. Get the full path name of the APK file for the desired package.

   adb shell pm path com.example.someapp
   

   This gives the output as: package:/data/app/com.example.someapp-2.apk.

2. Pull the APK file from the Android device to the development box.

   adb pull /data/app/com.example.someapp-2.apk
   

The location of APK after successful pulling will be at ../sdk/platform-tools/base.apk on your pc/laptop.