"Please press the button on the right."
In general, AのB usually is translated as A's B (トムの上着 -> Tom's jacket)... but as we can see here, that's not always the clearcut case with 右のバタン and others. It might be easier to think of AのB as "A is an element of B" since 右のバタン and バタンの右 can both be used. An example of the first would be the answer sentence here where it could be thought of as "the button that is on right", while the latter example would be "right of the button" and might be used as such in the following "バタンの右に名前を書いて下さい" (Please write your name to the right of the button). A bit confusing but hope that helped!