Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation).

The case where there is a reasonable suggestion, assumes that you work with some type of nonassociative algebras over a field, and all identities follow from the multilinear ones. In other words, the category of algebras you are studying is the category of algebras over some operad O. In this case, there is a nice way to describe a module over such an algebra. For an algebra A, a module structure on V is given by a collection of operations defined by all possible operations from O, where you are allowed to plug an element from V into one slot of operations, and plug elements of A into other slots. To write down the module axioms, take the defining identities of O, and form new identities, marking one element there in all possible ways; now treat the unmarked elements as elements of A, and marked elements as belonging to V.

For example, for associative algebras the original identity is (ab)c=a(bc), which leads to the following definition. A module structure is defined by two operations, a,v\mapsto av and a,v\mapsto va satisfying the identities (ab)v=a(bv), (av)b=a(vb), (va)b=v(ab). This means that in the case of associative algebras we defined bimodules. Also, for Lie algebras we get the module structure which, as it is immediate to check, coincides with the usual module/representation structure. In general, this construction provides a reasonable "enveloping algebra" for your nonassociative algebra. Thus, one way to approach your question is to study representations of the enveloping algebra, and sometimes it's the best you can get.

Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation).

The case where there is a reasonable suggestion, assumes that you work with some type of nonassociative algebras over a field, and all identities follow from the multilinear ones. In other words, the category of algebras you are studying is the category of algebras over some operad $O$. In this case, there is a nice way to describe a module over such an algebra. For an algebra $A$, a module structure on $V$ is given by a collection of operations defined by all possible operations from $O$, where you are allowed to plug an element from $V$ into one slot of operations, and plug elements of $A$ into other slots. To write down the module axioms, take the defining identities of $O$, and form new identities, marking one element there in all possible ways; now treat the unmarked elements as elements of $A$, and marked elements as belonging to $V$.

For example, for associative algebras the original identity is $(ab)c=a(bc)$, which leads to the following definition. A module structure is defined by two operations, $a,v\mapsto av$ and $a,v\mapsto va$ satisfying the identities $(ab)v=a(bv)$, $(av)b=a(vb)$, $(va)b=v(ab)$. This means that in the case of associative algebras we defined bimodules. Also, for Lie algebras we get the module structure which, as it is immediate to check, coincides with the usual module/representation structure. In general, this construction provides a reasonable "enveloping algebra" for your nonassociative algebra. Thus, one way to approach your question is to study representations of the enveloping algebra, and sometimes it's the best you can get.