In the strictest sense of the term, "numerical integration" means "quadrature". Quadrature is named after one of our culture's oldest problems, that of squaring the circle. This means, quite literally, producing a square that has the same area as a circle. Time proved this to be intractable with the Ancient Greek of compass-and-unmarked-rule geometry, but thanks to calculus we compute the area of any curve given to us in some useful manner. Sometimes quadrature can be done exactly, but in the general case numerical methods are needed. Therefore schemes such as the rectangle and trapezoid rules, Monte Carlo and Quasi Monte Carlo, etc.
However, most often "numerical integration" actually means "numerical schemes for differential equations". This is because differential equations are still one of the core tools (arguably the single most important) of mathematical modeling and because of the Fundamental Theorem of Calculus the following equations are identical:
(1a) d/dt x(t) = f(x(t),t)
(1b) x(t) = x(0) + Integrate f(x(s),s) ds from 0 to t.
That is: solving a differential equation is the same as calculating an integral. However, the most popular methods for quadrature (very importantly, Monte Carlo) are not commonly used, while there are specific methods for equations (1a,1b) that use differential equation structure that doesn't generalize to arbitrary integrals.
It' true that many of the more elementary numerical schemes for differential equations (particularly in the ODE family; PDEs, SDEs, RDEs are worlds of pain in their own) are derived from elementary quadrature methods. The rectangle method for quadrature, for example becomes Explicit (or Forward) Euler. However, schemes for differential equations are usually formed by methods taking advantage of the ordered nature of time, such as Taylor series and, in the real workd, Runge-Kutta schemes.
Moreover, conceiving of numerical schemes for differential equations as quadratures obscures the nature of the dynamical systems that we hope to model with differential equations. Chiefly: although a differential equation (1a) is given as the problem of discovering a map t → x(t), what we hope to know is the "law of motion" of the system, that is, the family of maps φt x(0) → x(t) that take states of the system to themselves. These are technically known as flow maps, and the point of conceiving a numerical scheme is to discover a discrete-time map ψn z(0) → z(nh) such that $z(t)$ sort of behaves like $x(t)$. This is what is meant by numerical integration basically anywhere in modern Applied Math research (and practice!).