order of a differential equation
Differential equations come in levels of complexity, and the simplest way to rank them is by asking: how many times do we have to differentiate the unknown? An equation that only mentions the first derivative is the gentlest kind. One that reaches into the second derivative is reaching into acceleration, into curvature. The order is just the label for how deep into the derivatives the equation goes.
Precisely, the order of a differential equation is the order of the HIGHEST derivative that appears in it. In y' = k y the highest derivative is y', so it is first order. In y'' + 3 y' + 2 y = 0 the highest is y'', so it is second order, even though y' and y also appear — only the top one counts. A general nth-order equation has y^(n) as its highest derivative. The order tells you, roughly, how much freedom its solutions have: a first-order equation typically needs one arbitrary constant, a second-order equation two, and an nth-order equation n.
Order is the first thing you check about any new equation, because it shapes everything that follows: how many initial conditions you must supply to pin down a unique solution, how many arbitrary constants the general solution carries, and which solving methods even apply. First and second order cover most of an introductory course, because most foundational laws of physics — Newton's law of motion among them — are second order in time.
y' + y = 0 is first order (highest derivative y'). y'' = -9.8 is second order (highest derivative y''), and indeed describes a falling body, since acceleration is a second derivative. (y')^2 + y = x is still first order — squaring y' raises the power, not the order.
Order = the highest derivative present. The power it is raised to does not change the order.
A common slip: confusing order (which derivative is highest) with degree (the power that highest derivative is raised to). (y'')^3 + y = 0 is second order, third degree — its order is still 2.