The Xbox One got an update on June 3rd which added, among other things, external storage. Unlike the Playstation 4's option to upgrade the console's internal storage that we covered previously, the Xbox One will be relying on external USB3 hard drives for expansion.
The Xbox One's SATA II internal storage supports 3GBit transfer speeds, which is slower than USB3, meaning that you'll be getting faster loading times by storing things on the external. That's a big win for One owners, who may have been disappointed they wouldn't be getting the loading time reductions Playstation 4 could enjoy by installing solid state drives.
Which drive you want depends on a number of factors, including how much storage you want and whether you'll be using this drive for everything or just leaving it attached to your One (we recommend the latter).
There are YouTube videos, forum posts, and Reddit threads like this one coming to similar consensuses about loading time increases if you'd like to read deeper into that.
Your easiest, cheapest bet here is going to be a 1 or 2TB portable external. That's going to triple or quintuple your storage without adding an additional power cord. We like the 1TB HGST Touro because of its 7200rpm speed and its price tag. The 1TB Western Digital My Passport that won Lifehacker's Hive-Five is a great drive, but it runs at 5400rpms, and we're trying to cut loading times here.
If you want to up the storage (you don't need it), then the WD My Book 4TB is the way to go, recommended in both the Lifehacker Hive-Five and by Wirecutter. The My Book is a very reliable, fast drive with a low price per terabyte.
Rolling Your Own
Your other option is grab an enclosure or docking station, like this one from Plugable (which is cool because it lays flat), and throw in a bare drive, whether it's one you buy for this purpose or one you happen to have laying around. You can even use an SSD with a setup like this, though you'll be seeing diminishing returns in loading times. This route can be cheaper if you really want that extra storage, and we like Western Digital's options for this purpose.
For a cheaper option, check out Seagate's offerings.
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