Superscript and subscript in LaTeX

x_1^2 + x_2^2

The order of ^ and _ does not matter: x_1^2 and x^2_1 give the same result. LaTeX stacks both indices above each other on the same letter.

a_{n+1} = 2^{10}

The braces mark the whole content of the index. Without them a_n+1 gives a with subscript n followed by an ordinary +1, which changes the meaning.

E_{\text{kin}}

\text sets the subscript upright, like ordinary text. That matters because kin is an abbreviation of a word, not a product of three variables.

e^{x^2}

Indices can be nested, and each level is smaller than the one before. With three or more levels the formula gets hard to read; write \exp(x^2) instead.

a_{ij} \quad a_{i,j} \quad a_{1,12}

With two one-letter indices a_{ij} is enough. A comma removes ambiguity once numbers appear: a_{112} could mean row 11 or row 1.

{}^{14}_{6}\mathrm{C}

The empty group {} gives the indices a base, so they land to the left of the symbol. This is how isotopes are written; for chemical formulas the mhchem package is more convenient.

c = 3 \cdot 10^{8}\,\mathrm{m/s}, \quad 10^{-3}, \quad x^{-1/2}

A minus sign counts as a character, so 10^-3 puts only the minus in the exponent. In a lab report with many such numbers, \num{3e8} from siunitx writes the power of 10 for you.

1\textsuperscript{st} place, H\textsubscript{2}O

Outside math mode ^ and _ are errors. \textsuperscript and \textsubscript keep the letters upright in the font of the paragraph, while $^{st}$ would set them in math italics.