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Understanding "logical OR" and "logical AND" in programming languages

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I'm posting this as a language-agnostic catch-all for a simple class of logical errors, so that beginner questions can be duplicated to it in the future. In my experience, the large majority of questions along these lines are asked by new Python programmers, but in principle the question applies to many languages.

This question is meant to be focused specifically on the theoretical understanding of the logical flaw - it is *not* about details of what the correct syntax looks like (only a language-agnostic description), and it is *especially* not about other ways to solve underlying problems.
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Many programming languages - most notably Python, but the question applies generally - either have keywords like `or` and `and` used for logic, or equivalent operators such as `||` or `&&` - which are referred to as "logical or" and "logical and" respectively in the language documentation. However, these tools don't seem to work in a natural or expected way for many beginners. Which is to say: "translating" an English sentence into code, using `or` or `||` to represent the English word "or", etc., commonly causes errors or gives the wrong result.

Why is this? What are the semantics of these operators, and how is that different from a natural-language understanding? And why are they called "logical"? (Are there other kinds?)