A theorem of Wedderburn says that finite division ring is field.
Here, ring means "ring with unity and is also associative".
In particular, finite division (associative) algebras are fields.
I was trying to get examples of following kind of algebras.
Q 1. What are the examples of finite division non-associative algebras over a field of order $p$, a prime?
Q.2 Where can I find the properties of algebras in Q. 1?
Note that there are simple examples of finite non-associative algebras, namely Lie algebras over a finite field of finite dimension; but they are not necessarily (perhaps never) division algebras.