I somewhat disagree with the purposes given so far, namely that this is to visually mark paragraphs.
While having such a visual distinction is a consequence of the indentation, it is not the purpose. Not justifying the last line would suffice for this (assuming justified text). The ambiguity in the rare case of a full last line can be avoided in historical manual typesetting by just slightly adjusting the justification gap in the paragraph or adding a respective penalty in to the TeX linebreaking algorithm. If it was only for this ambiguity, historical typesetters wouldn’t have introduced the indenting convention.
The main benefit of indenting paragraphs is that it gives the reader some visual reference points on the left side of the text which they can use when jumping from the end of one line to the beginning of the next. Without the indent, you have just a homogeneous wall of text and it happens more easily that you accidentally skip a line or read the same line twice.
There is no need for this at the beginning of a section (or similar), hence indenting is usually not used there. Also if paragraphs are spaced, this already provides visual reference points and obviates the need for indenting.
As for the popularity, everything I have ever encountered had either:
- indented paragraphs,
- vertically spaced paragraphs,
- no paragraphs at all (e.g., a brochure),
- strong typography and design flaws (overly long lines, bad colour contrast, inappropriate typeface), i.e., it was probably not professionally typeset.
And I am the kind of person who notices such things. (I read mostly German and English material.)