The quaternions are a noncommutative (and associative) division algebra.
Commutativity and aasociativity are totally independent, even for structures as rigid as algebras, and it would be a good use of your time to stockpile some more examples.
The octonions are a noncommutative nonassociative division algebra.
The real numbers are a commutative and associative division algebra.
The cross product on triples of elements from a field of characteristic 2 form a nonassociative but commutative algebra.
Of course, moving to groups and magma makes everything even easier. Groups are all associative, but of course there are Abelian ones and nonabelian ones.
Someone mentioned my favorite nonassociative commutative magma already: the rock paper scissors magma, where rr=r, pp=p, ss=s, rs=r, rp=p, ps=s.
I don't have a favorite nonassociative noncommutative magma, but I imagjne it is very easy to haphazardly define multiplication on a small number of symbols to make one. Say, a, b, c with ab=c and ba=b, ca=c and the other products however you wish.