Periodic motion is a motion that is repeated in equal intervals of time.
translatory motion is a motion in which all points of moving body move uniformly in the same line or direction.
oscillatory motion is a repetitive motion, typically in time,of some measure about a central value or between two or more differents states.

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A periodic motion is the movement of a particle in space that repeats in time.  The particle moves equal distances in equal periods of time.  The particle is at the same position after a duration called the period of the motion.

Example: uniform circular motion, oscillations of a pendulum, vibration of a diaphragm, movement of the needle in a stitching machine.  Planets going around the Sun is a periodic motion.

Oscillatory motion is a motion in which the particle moves to and fro in a straight line about a fixed point.  The center position is usually the equilibrium position and the distance to the extreme position and the center is called the amplitude of the motion.

Vibrations of a spring with a weight hanging from it is an oscillatory motion.

Translatory motion is the movement of the entire particle from one position to another position in space along a straight line.  It is actually along one direction or dimension. It can be a uniform linear motion or non linear uniform.

Example : movement of a car along a road in a lane