Actually, there is a terminology problem here. While all textbooks use the same definitions for "symmetric", "non-symmetric" and "anti-symmetric", the term "asymmetric" is ambiguous.
A relation is "symmetric", of course, if, whenever it has aRb, it also has bRa. An example would be on the relation on the set of people by "aRb if and only if a is a sibling of b". ("sibling"- brother or sister). If aRb then "a is a sibling of b" so it follows that "b is a sibling of a" so bRa.
A relation is non-symmetric if aRb does NOT imply bRa. Given aRb, bRa might be true but not necessarily. An example of a non symmetric relation is the relation on the set of people defined by "aRb if and only if a is b's brother". If aRb then "a is b's brother" and it might be the case that b is also male so "b is a's brother" and so bRa. But it might be the case that b is female and so is a's sister, not his brother. We would NOT have "bRa" in that case.
A relation is anti-symmetric if aRb implies that bRa is NOT true. That is, given aRb, we know that bRa is NOT true. An example of this might be the relation defined on set of people by "aRb if and only if a is male and b is a's sister". Now, if aRb, we know that b is female and so we cannot have bRa.
Notice that in "symmetric" if aRb then we must have bRa. In "anti-symmetric" if we have aRb then we cannot have bRa. In "non-symmetric" we are only saying "not symmetric". If aRb, we do not know if we have aRb or not.
Now the ambiguous part: many text books, I think most, use "asymmetric" to mean "not-symmetric" but a few use it to mean "anti-symmetric".
If this is for a class, I recommend you consult your text or teacher to see exactly how "asymmetric" is being used.