The Android OS is Open Source (formally known as Android Open Source Project AOSP http://source.android.com/ ), which means anyone can "fork" it into their own version of the OS. You can't use the word Android, the same font etc.; check the license: http://developer.android.com/legal.html
You can also combine your fork with other Open Source or proprietary software on a hardware device with the relevant Open Source and other licenses.
Android is a catch-all name for the OS which is really two parts: the "open" Android Open Source Project (AOSP), which forms the foundation of the OS, and the closed source proprietary services from Google, which are Google-branded apps, like Gmail, Maps, etc.
The use of Google services and APIs in conjunction with any fork or version of Android requires licensing from Google.
One example of a Android OS fork that does not use Google services is Amazon's Kindle Fire. Amazon used AOSP but developed their own app store, web browser, e-mail, etc.
You're free to use AOSP with your own Apps, own sets of functions and logos, as long as you include the relevant software licenses, as per the license linked above.
One legal caveat: if you are developing a device and OS fork and related software for sale, you should be sure and get a clear opinion from a attorney for your own records and protection and write a clear EULA to include with your software.