There are nominally two legal paths that Trump and his supporters can follow during the vote counting to overturn the election results. Neither one is likely, but they exist. (There's also nominally the possibility of a violent coup by military or paramilitary forces, but at that point all bets are off and anything is possible.)
The procedure for counting can be summarized as follows:
- Presiding officer (usually the VP) opens a state and reads off (or hands to someone to read) the certificate from the state's government with that state's electors' votes.
- Any objections to the validity of the certificate or any votes on it can be raised, provided that it is done in writing jointly by both a Representative and a Senator. Objections can be made that either the elector's vote was not "lawfully certified" (the elector shouldn't have been appointed) or not "regularly given" (something is wrong with the elector's vote itself).
- If there are no objections, go to the next state and start over with #1. If this was Wyoming (the last state), instead go to #7
- If there are objections, the two houses split up and debate for up to two hours.
- After debate, each house votes on whether the objection is valid.
- If both houses vote to consider the objection valid, that elector's vote (or the whole slate from the state, if they're all being considered together) is thrown out. If either house does not, it's considered valid and everything moves on with the next state.
- After all states' electors' votes have been counted, if one candidate has a majority of the "whole number of electors appointed", they win. If not, things get complicated.
The first path to overturn the results would be to invalidate enough electors for not being "lawfully certified" that Trump has more than Biden. Note that all of this is vaguely defined and subject to interpretation, but my understanding of the precedent is that if the vote is irregular it can be thrown out but the elector still counts as being appointed, and if the elector is not certified then the vote is thrown out and the total number appointed is reduced. (This distinction has never mattered before, so who knows?)
There are currently 538 electors and votes cast, so a majority is 270. Since Trump didn't reach that number, there is no way he can have a majority if the votes are thrown out but still counted as cast, but if 75 Biden electors are thrown out, then 232 > 231 and Trump wins. (I'll leave mathing out which combinations of states would have at least 75 Biden votes to the reader.)
The second path is to throw out enough votes that neither candidate has a majority. This only requires invalidating 36 Biden votes. This seems like it'd be harder, though, because the challenge is that the correct elector was appointed, but they then did something wrong (such as a faithless elector voting for a third party candidate). However, if this does occur, the House will vote - with one vote per state - for either Trump or Biden, and the majority state vote wins. Currently, Republicans control more state delegations than Democrats, even though the latter have the overall majority, so this would presumably be a win for Trump.
With the cooperation of a state governor or legislature in a Biden state, a competing slate of electors could have been appointed to vote for Trump, which would then provide a third path: Choosing the Trump-favoring slate when presented with two competing slates. But this didn't happen and it's now too late for it. Trump-favoring slates were submitted, but none of them have any official backing.
There's also been suggestions that the VP (who is presiding over all this) can invalidate or not count votes by himself instead of needing a Congressional objection, but there's no legal basis for that that I've seen, so I'm not even counting that as a fourth option.
Since the Democrats control the House and a significant number of Republican Senators have said they will oppose any objections, there is no realistic possibility of any of this actually changing the election results, but this is the theory of how it could happen.