Scalar
A scalar is a single real number with magnitude but no direction, used to scale vectors or represent undirected quantities.
Definition
A scalar is just a regular number, one with a size but no direction; temperature, mass, and distance are scalars, while velocity and force are vectors. Formally, a scalar is an element of the field $F$ (usually $\mathbb{R}$ or $\mathbb{C}$) used in a vector space; scalar multiplication $cv$ "scales" vector $v$ by factor $c$, stretching it if $|c| > 1$, shrinking it if $|c| < 1$, and reversing direction if $c < 0$. In a vector space $V$ over field $F$, the scalar field determines the structure, since $\mathbb{R}$-vector spaces and $\mathbb{C}$-vector spaces have different properties (every linear map on $\mathbb{C}^n$ has at least one eigenvalue, but this fails over $\mathbb{R}$); in physics, scalars are Lorentz-invariant quantities, unchanged by reference frame.
Example
Temperature ($72$ degrees) and wind speed ($20$ mph) are scalars, but wind velocity, $20$ mph northwest, is a vector because it has direction too. $3(2, 1) = (6, 3)$ stretches the vector by factor $3$, while $-1(2, 1) = (-2, -1)$ reverses it. The determinant of a matrix is a scalar encoding volume scaling, and the trace (sum of eigenvalues) is a scalar too; both are invariant under similarity transformations, and eigenvalues themselves are scalars in $Av = \lambda v$, where $A$ acts on eigenvector $v$ purely as scalar multiplication.
Key Insight
When you multiply a vector by a scalar, you change its length without changing its direction, unless the scalar is negative, which reverses it. In differential geometry, scalar fields are functions $M \to \mathbb{R}$ on a manifold, vector fields are sections of the tangent bundle, and the gradient of a scalar field is a vector field, connecting the two concepts.