What are translations, rotations, and reflections isometries?

The terms are related with mapping in mathematics. When you map a figure its dimension and direction changes. An enttire village can be mapped on a piece of plain paper. The distance between any two places in the village and  that on the map of the village are diffrent but bear the same proportion.

The postional shape of an object before and after a random movement,  can be treated as a mapping without change of dimension and shape of the object. But the object surely has undergone a displacement. But carefully note that  each and every identified points in the object may have different displacement from their corresponding positon in space before the displacement on account of  both translatory and rotatational  movement without our awareness.

Translation:  Imagine  a sraight line like stick in the position AB  maked with points A1,A2, A3 etc on it,move to the postion A'B' .Then if every marked point of the stick in the position A'B' bears the same distance from their corresponding postion before the move, then it is a translatory movement.

Rotation: If the  stick AB moves in a rotational movement then each of the marked points on the stick  moves in a fixed but dfifferent  circle. A1 moves in a fixed cirle. A2, the another  point moves in another cicle. Thus if A1 and A2 move in  same  plane  the two diffrent circular path of them are concentric.Otherwise, they may have the same axis of rotation.

Isometry:Both Rotation and Translation are isometric mapping.An isometric mapping is a mapping in which the relative distance between the two correpoding points in is maintained before and after the mapping. In other words The shapeand size  is maintained before and after mapping.

If you move a rigid body from position A to position B in space it gets displaced, involving both rotational and traslational movement. But any two marked point on or inside the object bear the same distance  after the movement. The property of retaining the same distane within itself after mapping is called isometry.

Hope this  non mathematical  and non pedantic  lay man's discription helps.