Based on what I understand, they are a generalization of prime ideals, and also are related to absolute values. How so?
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For every prime number, you can make an “absolute value” that is unique to that number. [Ostrowski’s Theorem](https://en.wikipedia.org/wiki/Ostrowski%27s_theorem) says that any absolute value is either the ordinary one *or* these prime-absolute values. If there were *only* prime-absolute values then we could just say that absolute values are primes. But there’s one more, the regular absolute value, so we can’t. We call them “places” instead.
We use the term “place” because we can construct similar kinds of absolute values on algebraic curves. There we actually see a similar patter, where all-but-one of these absolute values correspond to something nice and easy, in this case: Points on the curve. This extra point actually corresponds to the “point at infinity” for the curve. Interestingly, we can simply look at the curve differently to change where the “point at infinity” is and make a point at infinity look like any other point and, therefore, the corresponding absolute value shares the same properties. That is, the extra absolute value only looks differently because of how we decided to write down the curve.
Back in the case of numbers, it’s not so simple. The extra “place” *is* fundamentally different and we just can’t write numbers down differently to fix it. But all of this does suggest that the regular absolute value “should” correspond to some “prime at infinity”. There are actually many different frameworks out there to try and make sense of the notion of a “prime at infinity” which we only have access to because we have absolute values, or places. Ideally, we’d want a frame work where ALL of these absolute values pop out as special cases of a more general idea, but we haven’t yet figured out how.
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