To amplify WetSavannaAnimal's answer a bit, a mathematician defines a vector space (loosely) as a set of things that behave like little arrows when added together or multiplied by a scalar (aka number). They need not be little arrows. E.G. The set of all functions $y = ax^2 + bx + c$ is a 3D vector space.
An n dimensional vector can always be represented by n numbers, which is equivalent to a point in an n dimensional physical space, or a little arrow from the origin to that point. This is the sense in which a vector can be described by a magnitude and direction.
For the most familiar vector spaces, the numbers are real. But it is possible for them to be complex as well. E.G. the functions above could be defined over the complex plane. It would still be a 3D vector space. Even though $a$, $b$, and $c$ would be complex numbers, there are 3 of them.
This stretches the idea of an arrow in a physical space a bit. But then, so does a 4D or 17D vector. The point is that a scalar is the number that can multiply a vector without changing its direction.
To a physicist, a vector has to have another property. It must have a physically meaningful magnitude that does not change when you rotate the coordinate system. To a physicist, force is a vector, but a point in a thermodynamic phase space is not. To a physicist, 4D space-time is a vector space where the magnitude is the interval and the coordinate rotations are boosts.
Physicists are a bit sloppy on this point. To a mathematician, the idea of magnitude is captured by the definition of a norm. To a mathematician, 4D space-time is not a normed vector space because a norm must never be negative.
Getting back to the point, a second meaning of scalar is a physically meaningful value that is invariant under coordinate rotations. The magnitude of a vector is a scalar. Likewise, magnitudes of higher rank tensors are scalars.
In this sense, scalars are usually real numbers. Quantum mechanics has complex valued wave functions. But the physically meaningful magnitudes are real.