Unlike the storage drive in Sony's PlayStation 4, the 500GB hard drive that ships with the Xbox One isn't meant to be user-replaceable. That doesn't mean a drive swap isn't possible, however, just that it's a bit more difficult and could potentially void your warranty and get you banned from Xbox Live.
One individual decided to risk it all and swap out the stock Samsung hard drive for a hybrid solid state drive and an all-flash SSD. Fortunately for us, Brian Williams also documented the entire process in the video embedded below for everyone's viewing pleasure.
Specifically, Williams used a 500GB Samsung EVO SSD and a 1TB Seagate hybrid SSHD. A cold boot test reveals the solid state drive was able to boot to the dashboard roughly 4.4 seconds faster than the stock drive while the SSHD clocked in 2.1 seconds faster than the factory drive.
Another test with Call of Duty: Ghosts uncovered similar results with the stock drive loading the first level in 33.5 seconds. The SSHD did so in just 27.7 seconds while the pure solid state drive finished in 27 seconds flat. Unfortunately, Williams didn't perform any other game loading tests so we can't give it a complete comparison to the performance gains from an SSD swap in the PlayStation 4.
Aside from performance, one obvious reason to replace the stock drive is to add more storage capacity. But since Microsoft has already revealed support for external drives is in the pipeline, it might be best to just wait it out if that is your ultimately goal.