scalar field
Picture standing anywhere in a room and reading a thermometer: at every point there is one number, the temperature. A scalar field is exactly that — a rule that assigns a single number to each point of space (or of the plane). Pressure in the atmosphere, the brightness of a glowing wire, the altitude on a map, the electric potential around a charge: each is a scalar field, a quantity with magnitude but no direction.
Formally a scalar field is a function f(x, y, z) that takes a position and returns a real number. Its geometry is captured by its level sets — the surfaces (or curves) where f holds a constant value, like the isotherms on a weather map or the contour lines on a topographic chart. The field is smooth if you can differentiate it, and its rate and direction of steepest increase at each point are recorded by its gradient, the vector field grad f. So a scalar field naturally gives birth to a vector field.
Scalar fields are the starting raw material of vector calculus. Many of the most important vector fields in physics are gradients of a scalar — gravity from a gravitational potential, electrostatic force from the electric potential — and recognizing that a vector field is grad of some scalar (that it is conservative) is what makes a problem dramatically simpler. The operators of the subject (gradient, then divergence and curl on the resulting vector fields, and the Laplacian nabla^2 f built from them) all begin by acting on, or producing, scalar fields.
The field f(x, y, z) = 1/sqrt(x^2 + y^2 + z^2) is the electrostatic potential of a unit point charge at the origin; its level sets are spheres centered at the origin, and grad f points radially inward.
One number per point — and its contours and gradient already encode the physics.
A scalar field is not the same as a single number, nor a vector field with one component: it is a whole function of position, and its value at a point is independent of any coordinate system you choose, unlike the components of a vector.