Associativity with one operation or two (or more) operations It seems to me there are different 'types' of associative law that are all said to simply have the property of associativity.
For example this term is applied if we are only considering one operator e.g. $(x \circ y) \circ z = x \circ (y \circ z)$, or, as in the case of a vector space, two operators e.g. $\lambda (\mu v) = (\lambda \mu)v$ where $\lambda,\mu$ are scalars and $v$ is a vector. Presumably this would extend to three or more operators.
My question is: have there ever been different names for these seemingly different kinds of associative law?
For examples of this usage, please see here and here.
 A: If the question how the equation $\lambda(\mu v) = (\lambda \mu) v$ in the definition of a vector space is related to the usual notion of associativity:
The usual notion of associativity of a multiplication $*$ says that $L_x \circ L_y = L_{x*y}$, where $L_x$ is the operator which multiplies with $x$ on the left. But in case of a vector space, we also have, for every scalar $\lambda$, an operator (called homothety) which multiplies with $\lambda$ on the left, and we get the same equation.
Even more generally, let $X$ be a set and $M$ be a magma (a set equipped with a binary operation). Then an action of $M$ on $X$ is a function $M \times X \to X$, denoted by $(m,x) \mapsto m*x$, which satisfies $m*(n*x)=(m*n)*x$. This can be equivalently described as a homomorphism of magmas $M \to \mathrm{End}(X), m \mapsto L_m$.
Then the multiplication $M \times M \to M$ is an action if and only if the binary operation on $M$ is associative, i.e. $M$ is a semigroup. But also other actions are possible. If $M$ is a monoid there is a similar definition (requiring $1*x=x$), actually more generally for monoid objects in a monoidal category. For the monoidal category of abelian groups this precisely yields the notion of a module over a ring, in particular vector spaces.
Conclusion: The notion of an action is the most general incarnation of associativity.
A: Associativity is a property of a single binary operation.  I don't think I had seen the term 'associative' used for a relationship between multiple operations (but I have now from your comments below).  
I'm not sure if there's a general term for things like your example of $\lambda(\mu v) = (\lambda\mu)v$.  In that particular case, Wikipedia calls it 'compatibility of scalar multiplication with field multiplication'.
