Difference between a 'calculus' and an 'algebra' What is really the conceptual difference between a calculus and an algebra. 
Eg. Is SKI combinator calculus really a calculus?
A friend claims that free variables are fundamental for a calculus, and as such that SKI is not a calculus, but an algebra.
 A: Mathematics is an activity of investigation and exploration. Informally, both calculi and an algebras are tools which consist of sets of symbols and systems of rules (usually called axioms) for manipulating those symbols.  
Calculi tend to be specified/defined/explored/used to answer questions of "calculation" or reckoning, in some very general sense.  Calculi tend to be used to investigate properties of objects (i.e "What is the area under the curve?")
Algebras tend to be specified/defined/explored/used to answer questions about how different "things" are related, in some very general sense.  Algebras tend to be used to study the relationship between objects.  (i.e. "Is this equation 'the same' as that equation?")
I think it is safe to say that the term "algebra" today, carries a bit more meaning to most mathematicians than the general teram "calculus".  
As examples:
The Calculus (as taught in high-school or undergraduate university), also known as "infinitesimal calculus", is a calculus focused on limits, functions, derivatives, integrals, and infinite series. It is chiefly concerned with calculations or answering questions about change.  The Calculus uses the complex numbers (chiefly) as a foundation for this investigation.
Opening a book on computer science, you might find a "calculus of computation" which might involve symbols and rules which let one "calculate" or "discover" behavioral properties of a computer program.  As a foundation, such a calculus might use "states" and "transitions", instead of the complex numbers, to ground the investigation.
Elementary Algebra (ie. high-school algebra) is, informally, the study of relationships of variables and structures (e.g. equations) arising from combining variables according to certain rules (i.e. performing "operations").   It uses the complex numbers as the basic foundation in which one could "check" or "verify" statements, but quickly one finds that "calculating with numbers" is not that useful (or practical) in investigating relationships between equations. 
"The general theory of arithmetic operations is algebra: so we can also develop an algebra of set theory." - Concepts of Modern Mathematics, Ian Stewart
In that sense, Elementary Algebra is more "abstract" than arithmetic, and is often the subject where schools (specifically bad teachers) lose a student's interest and attention in mathematics.  It is a tragedy, since it is exactly at Elementary Algebra that things get interesting.  
In computer science or other engineering disciplines, you might find a "process algebra" when reasoning about how various states of a computer program relate to each other.  We can ask questions like "is a specification of a collection of processes 'functionally equivalent' to another specification (i.e do they do the same thing? as in the case of a particular hardware design versus a software program)?  The same "process algebra" could possibly be used to reason about how the various "states" of a garage door opener relate to each. Such an algebra might use states, transitions, and time as a foundation.
sigstop
A: As I vaguely feel it, and trying to follow the ethimology, calculus is linked to computing, while algebra is linked to solving. 
A: a calculus is a symbolic system for computation where computation can most generally be seen to be a spatial reorganization of symbols. any kind of numerical computation can be described in terms of recursion which can fundamentally be seen as a symbolic manipulation process. logic calculi and the calculus that set theory uses is also describable as a symbolic computation system.
an algebra is a mathematical structure in the informal sense which turns out to be a vector space with the added ability to multiply the actual vectors together. the complex numbers with addition and multiplication is an algebra over the complex numbers; http://en.wikipedia.org/wiki/Algebra_over_a_field 
A: In logic, the terminology seems to have been influenced by two factors.  The very early development of various deductive systems was done by people who were more philosophers than mathematicians and who seem to have used "calculus" to refer to anything that looked mathematical.  Also, that development took place before "algebra" had acquired all of its current meanings.  
My impression is that the use of "calculus" in logic is restricted to the meaning of "formal deductive system" --- and usually rather old systems.  As for the SKI system of combinators, I would call it a calculus if you're talking about rules of inference.  But if you mean the system of all combinators, with the operation of application, generated by S and K (I is redundant), then this is an algebra.
A: Webster's defines the primary definition of a calculus as follows:

a method of computation or calculation in a special notation (as of logic or symbolic logic)

Wiktionary gives a similar definition:

Any formal system in which symbolic expressions are manipulated according to fixed rules.

This agrees very much with the definitions I have encountered within mathematics. Free variables are not a requirement; indeed, even variables are not strictly required as objects within a calculus. (I am aware that there exist proof calculi that don't deal with the concept of variables; perhaps someone could give the name of such a one.)
A: In an algebra we would work within a known system and the results tend to be within those domains of interest and nothing new is found. In calculus the situation is different and we would encounter new things other than those we have been dealing. For example if we are working with one kind of curves and think about an operation like an integration or a differentiation we would be taken up/down another system. There is always a scope to see a new thing in a calculus and never a new thing in algebra. Whether it is a conventional mathematical system or relational algebra/calculus or logic (first order and more) we can observe this tendency.
