What are the acute differences between Direct and Indirect election?

Direct and indirect election are basically what they sound like. In a direct election, voters choose the candidate who wins. In an indirect election, voters choose one or more "electors" who then go on to choose the candidate.

In practice, this distinction can get fairly blurry. The election of the President of the United States is technically an indirect election, because of the Electoral College. Under the Constitution what we're actually doing is choosing electors who will then choose the President. But in practice, electors almost always choose the candidate who got the most votes in their own state (and people would get outraged if they didn't), so that for all practical intents and purposes we might as well have a direct election (albeit one with a weird counting system that more or less excludes all Republican votes in California and all Democrat votes in Texas).

A clearer example of indirect election is the Council of Europe, in which the Parliamentary Assembly is not elected by the voters in each country directly, but instead by the respective legislatures of the country who were in turn elected by the people.

Theoretically one could have higher-order indirect elections, in which you elect someone who elects someone who elects someone, but that very rarely happens.

Conversely, you could get even more direct than a direct election and go all the way to direct democracy, in which decisions are directly made by popular vote rather than by elected representatives. Switzerland does this quite often; it probably helps that the country is so small.

The differences between direct and indirect election can be quite important. The more direct the election, the more responsive the officials are to the people. This has upsides and downsides.

Some would argue that it's possible to be too responsive, and making elections direct simply makes politicians pander to a short-sighted public, while indirect elections allow policymakers to make wiser decisions on a longer time horizon. But personally I side much more with those who argue that the more direct we can make democracy the better, that being responsive to the public interest is what democracy is all about.