symmetry? you can think of it in many different ways. loosely speaking, it can be regarded as (an object) having a property when certain operation/action is done to the object, the object remain unchanged/invariant. A simple example is say a circle, you can rotate it by any angle in clockwise or anti-clockwise direction about its centre and it looks the same. So we say the circle has a rotational symmetry about certain axis. Precise definition of a symmetry can be defined by the language of group theory
which is itself motivated by these simple observations.
symmetry breaking? well... it means taking away those properties so that when the object is acted upon, the original object is no long the same as before the operation. A simple example: rotation by 120 deg about the centroid of an equilateral triangle is a symmetry. But we can break this symmetry by identifying that each of the three vertices as distinct
(eg. color them differently). If so, that rotational symmetry is broken because rotating by 120 deg no longer leaves the triangle looking the same. And now you need to rotate it by 360 deg or 0 deg etc. for it to be invariant.