Microsoft has been running an online profile system for a long time at this point - ever since the first generation of the Xbox console in-fact. However that does mean that if you were to make a new gamertag right now, you'd have to pick something that many tens of millions haven't already claimed.
To ease that up a little and make it so that people can have something a little more like what they want, instead of having to tack too many numbers and characters on the end, Microsoft is going to delete gamertags that were registered on the original Xbox and have not been used since.
Apparently this will result in around a million gamertags becoming available once again. Presumably a lot of these are quite simple too, standard names and references without addition.
The only downside is that Microsoft isn't going to tell us which ones it's freeing up. That does mean there won't be a mad scramble for specific names when the deletions go through, but you will have to guess to try and figure out what's available.
Gamertag deletions are said to have already gone through, with Microsoft opening up registrations for the newly available, old tags on Wednesday,May 18, beginning at 2PM EST.