The set of even numbers is a non-unital ring, in particular it has an associative multiplication, but you are right that it isn't an algebra.  For an example of a non-unital algebra, consider the set of continuous functions $\mathbb{R} \to \mathbb{R}$ which vanish at a particular point.  More generally, any proper ideal of an algebra is a non-unital algebra.

For an example of a non-associative algebra, take, for example, the <a href="http://en.wikipedia.org/wiki/Octonion">octonions</a>.  Your confusion arises because of the following issue: given a vector space $V$ and a bilinear operation $V \times V \to V$ we can associate to any $a \in V$ the linear operator $L_a$ which is multiplication by $a$, and these linear operators form an associative algebra.  However, composition of the operators $L_a$ need not be the same as $\times$: associativity is equivalent to the statement that $L_a L_b = L_{a \times b}$, which need not be the case in general.  I think this is the source of your confusion.