Bethesda confirmed today, via their livestream, that there will be a size limit for the number of mods you can download on Xbox One.
Xbox One players will only be able to download a maximum of 2GB of mods to use with their game. This isn't a bad limit, but it severely limits the scope for bigger, total-conversion type modifications, texture mods, and things of that nature. While 2GB isn't a terrible limit, it is still a limit.
Bethesda have also confirmed how mods are going to work on the console. A new "mods" menu item will be added, which will require you to link the game to a bethesda.net account before it becomes available. From here, you will be able to browse a catalogue of available modifications and choose the ones to install.
Much like the PC version, achievements are disabled when mods are enabled. Unlike the PC version, it seems unlikely that a mod to re-enable achievements will be available for the Xbox One version.
If a mod makes use of assets from any of the DLC packs, that mod will not be available to install unless you own that DLC.
Interestingly, when you install a mod, your save game is cloned, so if anything goes wrong, you can go back to a save without modifications enabled.
Mod support for Playstation 4 is also coming, and Bethesda confirmed they are on track to deliver this by June.