Paragraph mode is the most common; it's the one LaTeX is in when processing ordinary text. In that mode, LaTeX breaks your text into lines and breaks the lines into pages. LaTeX is in math mode when it's generating a mathematical formula. In LR mode, as in paragraph mode, LaTeX considers the output that it produces to be a string of words with spaces between them. However, unlike paragraph mode, LaTeX keeps going from left to right; it never starts a new line in LR mode. Even if you put a hundred words into an \mbox, LaTeX would keep typesetting them from left to right inside a single box, and then complain because the resulting box was too wide to fit on the line.
LaTeX is in LR mode when it starts making a box with an \mbox command. You can get it to enter a different mode inside the box - for example, you can make it enter math mode to put a formula in the box. There are also several text-producing commands and environments for making a box that put LaTeX in paragraph mode. The box make by one of these commands or environments will be called a parbox. When LaTeX is in paragraph mode while making a box, it is said to be in inner paragraph mode. Its normal paragraph mode, which it starts out in, is called outer paragraph mode.
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