Describe the purpose of regular and fair elections in a democracy. then explain how much u. s elections at the local, state, and federal level are similar in terms of district organization, how they are won, and the balloting system used.

First maybe we should define what a "regular and fair" election is.  Regular would mean that elections happen at regular intervals or according to a stated policy.  In other words, a government can't call off or postpone elections in order to preserve its hold on power.  "Fair" would mean all citizens have equal rights to participate in the democratic process.  That will mean civil liberties such as the freedom of speech, freedom of the press, freedom of assembly, etc, will be maintained.

Voting systems and ballot types used vary across the United States, but all systems and ballot types will be aiming to accomplish the same thing.  There are voter registration processes.  Absentee ballots and early voting options are often available.  Polling places are announced for each precinct of voters and sufficient voting hours are made available to give ample opportunities to all voters to participate.  For the process to be fully democratic, we want to do all we can to include every voter's ballot in the tally of votes.

How are elections won?  Well, in most cases a plurality of votes cast is enough to win.  In other words, you may not get a majority (above 50%), but if you have more votes than any other candidate you are still the winner.  Sometimes run-off elections are held if no candidate gets a majority of votes -- that typically happens in primary contests to seek a party's nomination as candidate for a particular office.

In the case of the US presidency, there is also the added layer of the Electoral College, which was a check-and-balance put in by our nation's founders against something they feared could happen by a "tyranny of the majority."  (I'll let you look up that term on your own.)