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A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, it is not a stable one: the ball can easily roll down the hill. At some point, the ball will spontaneously roll down the hill in one direction or another. The symmetry has been broken because the direction the ball rolled down in has now been singled out from other directions.
In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system will behave, can be split up into kinetic and potential terms
It is in this potential term (V(φ)) that the action of symmetry breaking occurs. An example of a potential is
as illustrated in the graph. This potential has many possible minimums (vacuum states) given by
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to φ = 0. In this state the Lagrangian has a U(1) symmetry. However, once it falls into a specific stable vacuum state (corresponding to a choice of θ) this symmetry will be lost or spontaneously broken.
In the Standard Model, spontaneous symmetry breaking is accomplished by using the Higgs boson and is responsible for the masses of the W and Z bosons.
In theories with scalar fields with an unstable potential, at low temperatures (i.e. near the vacuum state), the scalar field takes on a noninvariant background. It is precisely this noninvariant background which constitutes the spontaneous symmetry breaking; and not the vacuum per se. The symmetry breaking concept is certainly not restricted to the framework of explanation used above.