Solution
A solution is a value (or set of values) for the variable(s) that makes an equation or inequality true.
Definition
A solution is the value of the variable, or the ordered set of values for multi-variable equations, that makes an equation true when substituted back in; for an inequality the solution set is often an interval or union of intervals. A conditional equation has exactly one solution (or finitely many), a contradiction has none, and an identity has infinitely many, so recognizing which type you have before solving saves time. Formally, a solution to a polynomial equation $p(x) = 0$ is a root of $p$ in the domain of discourse; over $\mathbb{C}$ the solution set has size equal to $\deg(p)$ by the Fundamental Theorem of Algebra (with multiplicity), and for systems of polynomial equations, Bezout's theorem bounds the number of isolated solutions by the product of the degrees.
Example
For $x + 4 = 10$, the solution is $x = 6$, because $6 + 4 = 10$; always check by substituting the answer back in. For the system $x + y = 5$ and $x - y = 1$, the solution is the ordered pair $(3, 2)$, since $3 + 2 = 5$ and $3 - 2 = 1$. The equation $x^4 - 1 = 0$ has $4$ solutions in $\mathbb{C}$: $1$, $-1$, $i$, $-i$ (the fourth roots of unity), but only two real solutions, $1$ and $-1$.
Key Insight
Finding a solution is like finding the right key for a lock: only the correct value makes both sides of the equation match. The gap between the number of solutions over $\mathbb{R}$ and over $\mathbb{C}$ motivates the construction of algebraically closed fields, and the real-number solutions of a system correspond to the real points of an algebraic variety.