It's really impossible to keep this brief, but I'll try to keep it understandable:

The EC is a body of "electors", who serve as an intermediary body between the direct democracy of a popular national vote and the actual selection of a president. Their purpose is literally and intentionally to serve as a middleman, both to give a safety net to the ruling classes to make sure that whoever wins an election is someone they approve of, as well as to install a system that takes a national popular vote and basically applies an overlay to it...an overlay that leaves the process open to manipulation, stacking the odds, etc.

I'm not just saying this as a criticism of the system (though it is), this is the explicit purpose of the existence of the system.

Now to the nuts and bolts:

The US has a federal government with three branches: the executive (headed up by the president and including all of the various "Departments" like the departments of State (handling all diplomatic affairs), Defense (the military), Justice Department (FBI), Interior (National Park Service), Education, Agriculture, Homeland Security, etc.

Then there's the Judicial Branch, which is the federal court system, spearheaded by the Supreme Court. In addition to criminal trials involving federal crimes, they also have the responsibility of deciding on whether laws or actions of other government bodies are constitutional. If not, they have the authority to strike them down.

Last there's the legislative branch, which is responsible for creating laws and deciding how to spend money. Within the legislative branch, there are two bodies: the Senate, and the House of Representatives. This is because when the government was being created, states were much more independent than they are now, and there was a serious disagreement over how not only the people, but also the states would be represented in federal government.

So for the House, the number of Representatives each state sends is (roughly) proportional to that state's population; ie. a state with more people living in it will have more representatives than a state with fewer people living in it. The specifics have changed over time, and the way this system works is another issue, but that discussion is for another time.

However, smaller states, and (especially) states with slaves were concerned that even though they had a serious impact on the nation, they had a small voice in government. They wanted a system where their state was on equal footing with more populous states. Where just because they had less people (and by "people", in that time, they of course meant "land owning white male people"), they wouldn't have less power.

Thus there were two concessions given to these states to get them to join the union:

First, the three-fifths compromise: when determining population (to see how many representatives each state could send to the House), states were allowed to count each slave living in that state as three-fifths (0.6) of a person. Yes, these slaves, who their states regarded as property any other time, and who sure as hell weren't allowed to vote...were nonetheless to be allowed to count toward how much voting power their owners would have.

And second...the Senate. The Senate is the other house of Congress, where instead of determining members by population, it's much simpler: every state gets two. Regardless of population. This puts the smallest state on equal footing with the largest in the Senate.

And since both chambers of Congress (the Senate and the House) must pass a bill in order for it to become law, this is why it's so hard to get anything done for Congress.

SO!

Now that we know about the house and Senate and why and how they are the way they are... what's that have to do with the electoral college?

Well...the number of electors from each state are determined by adding up the number of Representatives and Senators that the state sends to Congress. So a small population state like say, Wyoming has one representative because very few people live there...and they get two senators because they are a state and all states get two. 2 + 1 = 3. So in a presidential election, Wyoming gets 3 electoral votes. For a more populous state, like my home state of Pennsylvania, we've got 17 representatives. Adding our two senators to that means that Pennsylvania gets 19 electoral votes for president.

Adding up all these electoral votes, it works out such that whichever candidate gets 270 electoral votes wins the presidency.

So you might be thinking, "Hmm... sounds like proportional voting and democracy with extra steps... what's the big deal?"

Well... there's two issues going on:

First: It's only proportional in allocation, but not so much in casting those votes. Of all 50 states, all but two (Maine and Nebraska) are set up such that whoever wins the state wins all of that state's electoral votes. So take my Pennsylvania for example: we've got about 13 million people living here. Obviously not everyone can vote, and not everyone that can vote will vote, but if next month, let's say all 13 million of us vote...if 12,999,999 people vote for Trump and 1 person votes for Harris, Trump wins all 19 votes. That makes sense. However, if Trump gets 6,500,001 votes and Harris gets 5,999,999 votes, that two vote difference means that Trump still gets all 19 votes. We don't split them so that he gets 10 and she gets 9. Winner take all.

Not only does this distort the popular vote, but it also has the effect of making a narrow victory in one area the same as a landslide in another.

Second: With the way votes are allocated, the fewest that any state can have is three (one representative and two senators). Even if ten people lived in that state, they still get three votes in the electoral college. Meanwhile, with the way congressional laws work, states with bigger populations do get more representatives...but as a state's population gets bigger and bigger, even though they get more electoral votes, each of those votes encompasses more and more people.

So looking (approximately) at Wyoming and California: Wyoming has a population of 582,000 and gets 3 votes, California has a population of 39,000,000 and gets 54 votes. That means that every vote in Wyoming represents about 194,000 residents, while every vote in California represents about 723,000 residents.

Doing the math, this means that every vote in Wyoming carries about 3.73x more power than a vote in California.

So in summary: the biggest criticisms of the electoral college are:

1. The lopsided way votes are allocated in the first place.

2. The winner-take-all system awarding the same number of votes for a landslide and a narrow victory distorting the actual voting numbers.

3. The lopsided allocation resulting in a situation where some Americans living in low population states having dramatically more power than others, based simply on where they live.

Of course these issues lead to lots of other weirdness and wrongness...for example: with the winner take all system, candidates don't even try to win states that are projected to safely go to one candidate or the other...they focus all attention on "battleground" states where the election is set to be close, ignoring millions of people nationwide because they happen to live in a state that's not competitive. A national popular vote would eliminate state political boundaries and make everyone's vote matter equally.

Likewise, this is how you end up with a case like 2016: more people voted for Hillary Clinton than Donald Trump...but those people lived in the wrong states, so basically she won by bigger margins but the margins meant nothing because he won narrow victories in more areas...so even though more people wanted her to be president, because of the electoral college, he got enough votes in the right geographical areas to win the presidency with fewer votes.