Motion with constraints applied is contrained motion.
Newtons's second law says: m a = F = m d^2 s / st^2 = m d^2 r / dt^2 = m dv/dt = dp/dt
where a, F, s , r v , p are all vectors with the usual meaning.
Here F is the net resultant force, that comes from the external forces as well as from constraints in the system and that result in the acceleration.
Suppose a particle has to move along a straight line. Then its component in the perpendicular direction is always 0. So there is a constraint that some Normal force cancels the component of resultant forces in that direction.
If the block is sliding along a slope, the normal reaction perpendicular to the incline balances the component of the gravity perpendicular to the slope. This is a constraint force. (normal reaction.)
If there are two or more masses connected by some ropes, and they are hung on the pulleys fixed to ceiling, then the motion of one mass is dependent on the motion of the other. The motion of one body is contrained by the weight of the other body.
Some constraints are that the rope, wire , string is inextensible and remains tight. So tension force is always same along them in either direction. Then the acceleration, velocities, displacement of the bodies tied to both ends will be the same.
When a spring expands or compresses, there is a restoration force, it is same at either end of the string.
A block which is on the top of a smooth hemispherical surface, slides along the surface. There is a constraint related to normal reaction. and it moves for some distance along the curve and then flies off. The block does not move vertically down under the influence of gravity.