What would happen to the republican party?
What would happen to the democratic party?
Probably nothing. But after this election, I think the time has come for the Republican party to gradually wither, and I'd then expect the Democratic party to split and form the two new parties.
(The US should really adopt proportional voting and a parliamentary system...)
What would happen to the US?
Mostly business as usual, but that will become more apparent during the rest of this election.
For all the screaming over Trump, the US governmental system is advanced enough that any one person can't hugely affect it. All the checks-and-balances will still be in place. And in any case, Trump is most likely far more moderate than his statements so far in the election has made him out to be.
What would happen to the world?
We'll laugh a bit, just as when Bush was (re)-elected. Then we'll continue on. Possibly Trump will come with some less-than-diplomatic statements, which will create lots of media headlines, and some tiny diplomatic incidents, but it'll all pass.
Really, Trump isn't anything particularly special. He talks less controlled than anyone else, but the US government will mostly carry on as before.