When LaTeX is processing your input text, it is always in one of three modes:
LaTeX changes mode only when it goes up or down a staircase to a different level, though not all level changes produce mode changes. Mode changes occur only when entering or leaving an environment, or when LaTeX is processing the argument of certain text-producing commands.
"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
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
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".