An algebra is a specific kind of structure which is itself formalized mathematically. I've never seen a formalization of the notion of "a calculus" and I believe it to be a looser term, as KC defined it.

Specifically, an algebra consists of a set (or several "sorts" of sets) and operations that reduce pairs of elements from that set (or the pairs can be triples, etc.) back into the set. Usually that set corresponds to the "semantics" of the algebra, and syntactic equations like xy = yx exist in a different realm from the operations and their actions. 

Lambda calculus differs from an algebra by having a construct (lambda abstraction) that only makes sense if you know the syntactic structure of the term it applies to. That is, it has a binding construct. You could define lambda calculus as an algebra by taking the underlying set to be the syntax of the calculus itself, but that would require infinitely many operations (a lambda-binder for each variable) and equations, so perhaps that would be awkward.

Pi calculus, like lambda calculus, has binders, while "process algebras" are usually defined via operations on processes. I believe this to be a general trait of things described as "calculi"--that they have some form of name-binders, but I have never seen that observation written down.

I'm sure that an algebraist could give a more definite answer about this.