Coefficient
In a chemical equation, a coefficient is the full-size number placed before a formula that tells how many molecules or moles of that substance react.
Definition
A coefficient is the big number written in front of a chemical formula, and it tells you how many of that whole molecule there are. In an equation it is a whole number giving the relative number of molecules, formula units, or moles taking part; coefficients are adjusted to balance an equation so that atoms are conserved, a coefficient of $1$ is understood and not written, and you multiply the coefficient by each subscript to count atoms. Stoichiometric coefficients express the molar ratios in which reactants are consumed and products formed, so they are the numbers used in mole-to-mole conversions and limiting reactant calculations. The set of coefficients in a balanced equation is unique up to a common factor and is conventionally reduced to the smallest whole numbers, and they also appear as exponents in equilibrium expressions and as multipliers of molar enthalpies in thermochemical equations.
Example
$2H_2O$ means two water molecules, which is $4$ hydrogen atoms and $2$ oxygen atoms in total. In $2H_2 + O_2 \rightarrow 2H_2O$, the coefficients $2$, $1$, and $2$ mean two hydrogen molecules react with one oxygen molecule to make two water molecules; the left side has $4$ H and $2$ O, the right side has $4$ H and $2$ O, so it is balanced. For $CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$, burning $0.50\text{ mol}$ of methane requires $0.50 \times 2 = 1.0\text{ mol}$ of oxygen and produces $1.0\text{ mol}$ of water, and since $\Delta H = -890\text{ kJ}$ per mole of methane, this releases $445\text{ kJ}$.
Key Insight
A coefficient multiplies everything in the formula after it, while a subscript only counts one element. To balance an equation you may only change coefficients, never subscripts, because writing $H_2O_2$ to balance oxygen would change water into hydrogen peroxide, a different substance. Coefficients are ratios, not absolute amounts: a recipe calling for $2$ eggs per $1$ cup of flour scales the same way, and the "limiting reactant" is just the ingredient you run out of first.