Time Travelers (miraijin, literally "future person") in the Haruhi Suzumiya franchise are humans who travel though time to complete a certain quest or to ensure that time flows the way it is supposed to, through the use of a TPDD.
Every time time travel is used with a TPDD, it causes damage to the time plane it punches a hole through. Longer "distances" cause more damage. The primary role of time traveling to the past is to repair this damage, otherwise the effects will propagate into the future. Time travelers will be aware of these changes even if no one else is.
Time travel cannot go back beyond year -3 due to a "timequake" generated by Haruhi Suzumiya.
There are at least two factions of time travelers, one of which, represented by Mikuru Asahina, works with Haruhi Suzumiya, and another represented by Fujiwara, works with Sasaki. Asahina's faction intends to protect her future, while Fujiwara's faction hates being unable to change it. His faction even tried to kill one of the inventors of time travel, the Bespectacled Boy (aka Hakase).
Later Fujiwara tried to hijack Haruhi's special powers in order to transfer them to a more suitable vessel. He proposed Sasaki as this vessel, but revealed he didn't really care who the power went to. He intended to use such power to alter time beyond year -3 in such a way that Mikuru Asahina would not recall changes to the timeline.