divergence
Stand at one point inside a flowing fluid and ask a simple question: is more stuff coming out of a tiny region around me than going in? If the fluid is being created here (a tap, a source) the answer is yes; if it is draining away (a sink) the answer is negative; if whatever flows in also flows out, the answer is zero. Divergence is the number that answers this question at every point of a vector field — the local rate at which the field spreads outward.
For a vector field F = (P, Q, R), the divergence is the scalar field div F = nabla dot F = partial P / partial x + partial Q / partial y + partial R / partial z. It is the dot product of the del operator with F, and the output is a scalar (a number at each point), not a vector. The honest definition is a limit: div F at a point is the outward flux through a tiny closed surface around the point, divided by the volume it encloses, as that volume shrinks to zero. So divergence is flux-per-unit-volume — the source density of the field. A field with div F = 0 everywhere is called solenoidal or incompressible: it has no sources or sinks.
Divergence is one half of the structural backbone of electromagnetism and fluid mechanics. In Maxwell's equations, div E = rho/epsilon_0 says electric charge is the source of the electric field, and div B = 0 says there are no magnetic monopoles. In fluid flow, div v = 0 is the statement of incompressibility. The divergence theorem then lifts this pointwise source density up to total flux through a whole closed surface, which is why divergence and flux are two sides of one idea.
For the outward radial field F = (x, y, z), div F = 1 + 1 + 1 = 3 everywhere — the field spreads uniformly, as if every point were a small source. For the rotation F = (-y, x, 0), div F = 0: pure swirl creates and destroys nothing.
Spreading gives positive divergence; pure rotation gives zero — divergence sees sources, not swirl.
Divergence of a vector field is a scalar; do not confuse it with curl (a vector) or with the gradient (which acts on a scalar to make a vector). A common slip is to think a field with field lines that curve must have nonzero divergence — curvature of the lines is about curl, while divergence is purely about whether their density grows or shrinks.