A relation R is antisymmetric if,

 for all x and y,  x R y and y R x  =>  x == y.

I.e. no two different elements are mutually related.

Partial orders and total orders are antisymmetric. If R is also symmetric, i.e.

x R y => y R x

then

x R y => x == y

I.e. different elements are not related.