It simply means that there is a 50% chance that Peter has a brother. Peter's sex is not given and it has 50% chance that it may be a girl.
It is given. "At least one of whom is a boy" means that the number of boys is guaranteed to be greater than 0. Therefore, one child is a boy, and I pinpointed him as being the given of the problem, and gave him a name to differentiate him from the other unspecified child. If the problem had stated "Exactly one of whom is a boy", then the probability of both children being boys is 0 (0%), because the number of boys is guaranteed to be greater than 0 and less than 2.
Assuming that all families have exactly 2 children with random sex distribution:
1. Is at least one of your children a boy? - Yes, it is. Then the possibility that your other child is a son is 1/3.
Because 75% of families have at least one boy and 25% have two boys.
You gave the right probability here, but for the wrong reason. "Is at least one of your children a boy?" answered in the affirmative means that you can now answer the question as if one boy was a given. The question now is, "Given that I have at least one son, what is the probability that I have 2 sons?"
Per this page, this can be written as P(2 sons | at least 1 son) = P(2 sons and at least 1 son) / P(at least 1 son) = (1/4) / (3/4) = 1/3.
And I just invalidated all of my other comments on this thread... Ouch!
*takes a huge bite of humble pie*