If a system has symmetry, it means that something stays the same under a well-defined set of changes. Many constructs in nature are the result of such symmetry being broken.
Imagine a ball sitting balanced at the very peak of a hill. The system is symmetrical because if the hill were rotated about the axis running down through its top (or if an observer changed their viewing angle) it would still appear the same. It has rotational invariance or symmetry. Such a configuration is not stable, ultimately the ball will roll down from the top of the hill.
When the ball does roll down, the rotational symmetry is broken. The ball is no longer pierced by the axis of rotation and so the system looks different if rotated. As no outside force explicitly destroyed the symmetry, this is sometimes described as spontaneous symmetry breaking.
The actual direction the ball rolled off in is (if the hill is perfectly smooth and even) arbitrary, one direction would be as good as another. But, once a particular direction is chosen, we are stuck with it. Other possible directions, even if comparable to that in which the ball actually rolled, become nothing but "the road not taken".
Well, for federated identity standards, the ball has rolled down the hill and the direction it went is SAML 2. What might have previously been argued as a symmetrical state of affairs between SAML and other proposals (in terms of functionality, market uptake etc) is incontrovertibly broken (the symmetry not the standard) now. Perhaps not spontaneous, but broken all the same.
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