Balanced Forces
Balanced forces are equal in size and opposite in direction, so they cancel out and produce no change in an object's motion.
Definition
Balanced forces are forces that cancel each other out: they are the same size but push or pull in opposite directions, so the object does not change how it is moving. Forces are balanced when the net force is zero, and an object under balanced forces stays at rest if it was at rest or keeps moving at constant velocity if it was moving; balanced forces cannot cause acceleration. Formally they satisfy $\sum \vec{F} = 0$, the condition for translational equilibrium, which in components means $\sum F_x = 0$ and $\sum F_y = 0$ at the same time. Equilibrium can be static (at rest) or dynamic (constant velocity), and Newton's first law makes no distinction between them. For an extended object, full equilibrium also requires $\sum \vec{\tau} = 0$.
Example
In a tug-of-war where both teams pull with the same strength, the rope does not move because the forces are balanced. A parachutist falling at a steady $5\ \text{m/s}$ has gravity pulling down with $700\ \text{N}$ and air resistance pushing up with $700\ \text{N}$, so the velocity stays constant. For a two-dimensional case, a $20\ \text{kg}$ sign hangs from two cables, each at $30^\circ$ above horizontal. Vertical balance requires $2T\sin 30^\circ = 196\ \text{N}$, so $T = 196\ \text{N}$ in each cable, and the horizontal components, $T\cos 30^\circ$ each way, cancel.
Key Insight
Balanced forces do not mean no forces. A book on a table has gravity pulling down and the table pushing up; both are real, they just cancel. A car cruising at a steady $100\ \text{km/h}$ on a highway has balanced forces too, with the engine's forward push exactly matching air drag and friction, so constant velocity always means balanced forces. One distinction matters: balanced forces on an object are different from action-reaction pairs. The two balanced forces act on the same object, while an action-reaction pair acts on two different objects and never cancels.