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