Coefficient
A coefficient is the numerical factor multiplied by the variable in an algebraic term, such as 5 in the term 5x.
Definition
A coefficient is the number written right in front of a variable, telling you how many of that variable you have; it is the numerical factor that multiplies the variable part of a term. A term like $x$ has an implied coefficient of $1$ (since $1$ times anything is itself), and $-x$ has an implied coefficient of $-1$. In a polynomial $p(x) = a_nx^n + \ldots + a_1x + a_0$, each $a_i$ is the coefficient of the corresponding power of $x$, and coefficients belong to a ring or field such as the integers, rationals, or reals; in linear algebra they also appear in linear combinations like $v = c_1v_1 + c_2v_2 + \ldots + c_nv_n$.
Example
In $7x$, the coefficient is $7$, meaning $7$ groups of $x$. In $6x^2 - 4x + 9$, the coefficient of $x^2$ is $6$ and the coefficient of $x$ is $-4$ (the $9$ is the constant term, not a coefficient). In the polynomial $3x^3 - x^2 + 5$, the leading coefficient is $3$ (for $x^3$), the coefficient of $x^2$ is $-1$, the coefficient of $x$ is $0$, and the constant term is $5$.
Key Insight
Coefficients determine the "weight" of each variable term: when combining like terms you add or subtract coefficients while keeping the variable part unchanged. The leading coefficient of a polynomial also determines its end behavior and scaling, playing a central role in factoring, the Rational Root Theorem, and polynomial division.