The Time Stream is a portal on an unnamed planet. Every year, on New Year's Eve, a new member of the Spirits of the Years Past enters the Time Stream in order to "guide the new year on its intended course". The one to enter the Time Stream must be the one that represents the coming year; should a past spirit enter, time will fold in on itself and it would be the end of time.
In Sonic the Comic, Amy Rose and Tekno the Canary arrived near the Time Stream in time for 1999 to enter the stream. Unfortunately, 1930 had gone crazy and wished for time to be restored to how he remembers it, threatening all of time and summoning the Time Waster. Before the ghost could achieve his task, Amy convinced him not to and 1999 was free to enter as usual.
It is isn't explained what happens to the new spirit after they enter the Time Stream. At some point before the next New Year, the last spirit to enter the stream must somehow return to the spirits' world, as 1998 was already there, despite having enter the Time Stream almost twelve months ago.