Equation
An equation is a mathematical statement that two expressions are equal, indicated by an equals sign.
Definition
An equation is a math statement saying that two expressions are equal, always marked by an equals sign ($=$) in the middle; it may involve one or more variables, and it is true only for the specific values of the variable(s) called solutions. Not all equations behave the same way: some have no solution (contradictions like $0 = 5$), some have infinitely many (identities like $2x = 2x$), and most have exactly one (conditional equations). Formally, an equation is a formula of the form $P = Q$ where $P$ and $Q$ are expressions over some domain; solving an equation over a ring $R$ means finding elements of $R$ satisfying the relation, and polynomial equations of degree $n$ over algebraically closed fields have exactly $n$ roots, counted with multiplicity, by the Fundamental Theorem of Algebra.
Example
$3x + 5 = 14$ is an equation saying that $3x + 5$ and $14$ are the same value; solving it gives $x = 3$. For $2x - 7 = 9$: add $7$ to both sides to get $2x = 16$, then divide by $2$ to get $x = 8$, and checking confirms $2(8) - 7 = 9$. The equation $x^2 - 5x + 6 = 0$ factors as $(x-2)(x-3) = 0$, giving roots $x = 2$ and $x = 3$ over $\mathbb{R}$; over $\mathbb{C}$, every degree-$n$ polynomial equation has exactly $n$ roots.
Key Insight
Think of an equation as a balance scale: whatever is on the left must equal whatever is on the right, and your job when solving is to keep the scale balanced while finding the unknown. The Fundamental Theorem of Algebra guarantees the completeness of $\mathbb{C}$ as a root field, which is historically why complex numbers were introduced: to ensure polynomial equations always have solutions.