a first-order system

Real things rarely change one at a time. The number of rabbits depends on the number of foxes, which depends on the rabbits; the voltage on one capacitor in a circuit pushes current that changes another. When several quantities all change together, each one's rate of change depending on the whole set, you need not one equation but a team of them, all coupled. That team is a first-order system.

Concretely, you have several unknown functions of the same variable t — call them x1(t), x2(t), ..., xn(t) — and one equation for each derivative: x1' = f1(t, x1, ..., xn), x2' = f2(t, x1, ..., xn), and so on. 'First-order' means only first derivatives appear: no x'' anywhere. The unknowns are stacked into a single vector x(t) = (x1, ..., xn), and the whole team is written compactly as x' = f(t, x). Each f_i may freely mention any of the other unknowns — that mutual dependence is exactly what 'coupled' means, and exactly what makes the system more than n separate problems.

First-order systems matter because they are the universal currency of differential equations. Any single higher-order equation can be rewritten as a first-order system, and almost every numerical solver — Euler's method, Runge-Kutta — is built to march a first-order system forward in time. So learning to think in systems is not an extra topic; it is the form into which nearly everything else gets translated.