On November 16th, Microsoft added the ability to remap the buttons on your standard wireless controller for your XBox One. Those who want to change their button mapping can either launch the Xbox Accessories app or enter their Xbox One settings. If you don't have the app already, the link above will provide all the information you need to get it.
Remapping through the accessories app works by either swapping buttons or changing individual buttons through drop down menus. Looks altogether pretty simple. Those settings will only save on your console however, as the standard controllers don't have any on-board storage. So if you map your controls at home then take your controller to a friend's place, don't expect them to still be mapped the way you want. And sadly as of yet there is no PC support for said remapping, but it's on the way according to Microsoft.
[All Image Credits, Microsoft]