The first trip we see is made by Tim on New Years Day, immediately upon being informed by his father that he can travel to his own past. He chooses to travel to the party of the previous night, and being there with foreknowledge of how he made a fool of himself at the party he adroitly avoids doing so the second time through. This becomes our first real information about how it works and what happens.
It should be noted that to do this, Tim enters the hall closet, makes the jump, and exits the from the hall closet into the party; when he has finished his visit, he re-enters the closet and returns to the same closet in his own time. Some of those details will be important later; what matters for now is that he leapt to the past and returned to the future, and that when he did so he altered his own history such that he never made a fool of himself at the party the night before. We, though, must trace that history.
In the original history, Tim made a fool of himself at the party, and still regretted it the next day. Then his father told him that he could travel to his own past and alter it, and although he did not believe it, his test scenario was to undo the embarrassment, at which he succeeded. He then skips the intervening events and returns to New Years Day.
However, he does not cease to exist in the past. Rather, the version of him for which it is still New Years Eve apparently abruptly finds himself in a closet with no knowledge of what has happened for the past ten minutes, the very ten minutes when the countdown reached zero and the new year began and everyone celebrated. He did not embarrass himself, but he probably decided something was wrong with him. The next morning he is not thinking about how he embarrassed himself the night before, because neither he nor anyone else remembers him having done so. His father tells him of his unsuspected ability, and if he has any reason to go back to the party it is to find out why he had that blackout, and what he was doing at that time. He cannot, however, undo the cause of his embarrassment, because he is unaware of it--although probably he would be less enthusiastic about the New Year and more circumspect in trying to observe the events around him, so probably he would not embarrass himself if he made the same trip.
The complication is that it is not very likely that he would do so. There are probably hundreds of times he might prefer to revisit, and the specific impetus for him to travel to last night has been erased. That means there is no reason for him to make that trip, and so he does not make it (he makes some other trip instead), and thus his doppelganger, not having the benefit of foreknowledge, stumbles into the embarrassment he had undone. We are probably at this point trapped in an infinity loop, and that means the movie has ended. Of course, as we noted, there have been hundreds of such trips made by his relatives, most of which will have suffered the same fate as this one, and so we cannot have reached the present.
One way to save this is to use the common understanding of Niven's Law, and so to suppose that once a time traveler has altered the past he need not in the new version of events make the trip to alter the past, because once he has changed it it remains changed. Objections to this view have been made elsewhere, but if we allow for the moment that in this film that rule applies it becomes an issue to which we will have to return later.
There is another significant issue with this particular alteration to the past that we will have to consider next time, pertaining to Tim's memories.