In some informal notes that are not publicly available and for which I do not have permission to reproduce here, there is a reference to a 'finite-dimensional self-dual algebra over a field $K$'.
I can't find much about this, so I need to clarify what is most likely meant by it. It is discussed in the context of homological algebra and I assume (from the context) that it is meant to be a self-injective ring.
My best guess is that it relates to the connection between algebras and coalgebras rather than anything to do with duals of vector spaces. I haven't checked all the details yet but there appears to be a functor from the category of unital $K$-coalgebras to the category of unital $K$-algebras (not necessarily commutative) and 'self-duality' means that this functor is an isomorphism, so that there is a one-to-one correspondence between algebras and coalgebras. Does this sound right?