Democracy comes from the ancient Greek words "demos" and "kratos", together meaning "people power". If you have a governmental system where the people have some say in who governs, or may even sometimes get asked what they think on specific questions. you have a democracy.
In ancient Athens, the city was small enough for all male adult free citizens to assemble from time to time to vote on things. (Of course slaves and women didn't count!) That would be a pure democracy and there are still two small parts of Switzerland where they do this once a year to vote on any laws they like and choose the cantonal council for the next year. But anywhere else is just too big to have a big meeting of all citizens, so we go for a representative democracy instead - elect representatives for the next few years to do the governing and making laws for us. And if we don't like what they do, there will be an election sometime soon and we can vote for better representatives then.
It is also important in a democracy that when we vote, there is a real choice and we can vote freely. Communist countries have elections to LOOK democratic, but when there's only one choice, what use is the election? North Korea has the biggest turnout of voters in the world and the most one-sided results, because the secret police want to know why if you don't vote or you don't vote for Kim. And what if the counting is fiddled? That's how Mugabe always won in Zimbabwe.
So you want:
- a secret ballot so how you vote remains between you and the ballot box or the voting machine
- a real choice of who or what to vote for
- assurance that how it is all counted is foolproof, and that's why it's good to see if other countries send observers. They can see if people are being intimidated when they go to vote and that the count is being done fair and square.
Incidentally, this is why I like that in the UK, we vote on paper and it's all counted by hand. Nothing is hidden in any kind of machine, and the candidates and their representatives can wander around and see the count being done. What happens is the local council will hire the biggest hall in the area (mine uses the local university sports hall) and it all gets done overnight. Which makes wonderful TV for politically-minded insomniacs. If there is any complaint, any candidate can demand a recount and this often happens in a close result. Recounts are quicker because the papers are already in bundles of 100 that should be all for the same candidate, so all the counters have to do is check that no paper got into the wrong bundle and total it all up again.
The big objection to democracy is that it is "rule of the mob" and can we be sure that the mob knows what's best? Plato explored this over 2000 years ago in his book "The Republic" and he said that government should be in the hands of people trained for the purpose. But how do you ensure that they are not corrupted, like a king could be? How do you train these philosopher-kings and who should do it so they have the right ideas?
So we go for democracy because the alternatives have problems too, and it is democracies that have lasted longest and actually work as well as anything. Winston Churchill said that "many systems have been tried in this world of sin and woe" and he concluded that "democracy is the worst system in the world except for all the others that have been tried from time to time". He was no dummy and at least we can say of some kind of democracy is that it works and gives us all a say.
The USA is best described as a democratic republic. Republic because there is no king - that's the easy bit. (Being a republic has nothing to do with the rule of law, as so many Americans seem to have been taught. Countries with non-royal dictators who do as they please are republics.) Originally the US constitution provided for only the House of Representatives to be democratically elected. The Senate was chosen by the states because it is supposed to represent the states, and the President is elected by electors chosen by the states The constitution says nothing about how the states do their choosing. But normally the state legislature, elected by the people, would do it.
The USA has become more directly democratic since. There was an Amendment in 1912 that made the Senate directly elected by the people. And since 1868, all states have relied on having a vote of the people to choose their electors. But still, none of the states choose their electors in proportion to the vote - it's nearly always "winner takes all", where whoever gets most votes gets ALL their slate of electors chosen, even if it's only a win by one vote.
Which brings us to Trump. If the president had been elected by direct popular vote, Hillary Clinton would have been president for the last 2 and a bit years. But the system isn't that and Trump won more votes in the right places than she did, so he won out in this two-stage election.
As a European, I can certainly say we have different kinds of voting systems. The UK elects its House of Commons in the same way as the USA elects its House of Representatives - divide the country into voting districts and whoever gets most votes in the district gets elected to the House. But this means little parties get shut out. We regularly used to have the Liberal Democrats getting 25% of the vote but hardly anyone elected. Most other European countries use some kind of proportional representation, which means little parties will get people into the national legislature too and they can have some influence. It forces parties to work together to get things done. Is that better than just having two big parties who are the only ones who ever get elected? Very good question and that's one for you to think about.