I would like to clarify some definitions.
Let $F$ be a field.
1) An $F$-algebra is a vector space $V$ over $F$ with a bilinear multiplication $V\times V\to V$.
2) Let $A$ be a not necessarily commutative ring. An $A$-algebra $B$ is given by a ring homomorphism $\phi:A\to Z(B)$, where $Z(B)$ is the center of the ring $B$. In particular, we have $A\times B\to B$ given by $(a,b)\mapsto \phi(a)b$.
Now, in the latter definition, it seems they sometimes simply take $\phi:A\to B$ rather than $\phi:A\to Z(B)$. I think they do this only when $A$ is a commutative ring. I think having this ring map with codomain $Z(B)$ is necessary, else the multiplication from definition 1 wouldn't be bilinear.
Am I correct that these definitions 1) and 2) are entirely equivalent, as long as we do work over a commutative ring $F=A$? In such a case, I can see the bilinearity holding, since for example $(av,w)=(v,aw)$ since $\phi(a)vw=v\phi(a)w$. But in the case that $A$ is not commutative, the bilinearity would fail.
For $A$ commutative, definition 2 describes the special case of a unitial associative algebra under definition 1, assuming that your rings are defined to be unital and associative. Note that definition 1 works fine if you replace $F$ with $A$ and "vector space" with "module".
Note that definition 2 does not work if you drop the unital condition from your definition of a ring. For example, there is no "canonical" map from $F$ to a Lie algebra over $F$.
As Ben Webster points out here, there isn't really a single "correct" generalization of the notion of an algebra to noncommutative $A$.