Algebras for monads in Cat and 2-categories An algebra for a monad $(T, \mu, \eta)$ on a category $\mathbb{C}$ is defined as a morphism $T X \to X$ for some object $X$ such that the obvious diagrams commute.
If I look at the monad as a monoid in the category of endofunctors, actions for a monad seem to be a generalisation of monad algebras: an action for $T$ is a natural transformation $T \circ F \to F$ for some functor $F$. So a monad algebra is an action for which the underlying functor is constant.
Is there a reason why we can concentrate on constant functors when studying monad algebras instead of studying actions in general?
 A: So, you know that if you have a monoid $m$ in a monoidal category $(M, \otimes)$ then it's a natural thing to do to look at actions of $m$, namely morphisms $m \otimes c \to c$ satisfying etc. where $c$ is another object in $M$. One way to motivate the definition of monads is that you can actually do something more general than this: $c$ need not be an object of $M$! Instead, if you specify an action of $M$ on a category $C$, or equivalently a monoidal functor
$$M \to \text{End}(C)$$
then you can make sense of what it means to specify an action of $m$ on an object $c \in C$, where instead of $m \otimes c$ you use the action of $M$ on $C$. This is an instance of the microcosm principle. It is possible to describe, for example, algebras over an operad in this language, where $M$ is the monoidal category of species under composition.
The question then arises: what is the most general kind of monoid that can act in the above sense on an object in $C$? The answer is a monoid in $\text{End}(C)$, or equivalently a monad on $C$. So the reason that we think of monads as acting on objects in $C$ and not on objects in $\text{End}(C)$ is that $C$ is where our focus is. We want to understand $C$ and maybe categories monadic over $C$; we aren't really trying to understand monads for their own sake. 
