Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc.
Computer algebra is widely used to experiment in mathematics and to design the formulas that are used in numerical programs. It is also used for complete scientific computations, when purely numerical methods fail, as in public key cryptography or for some non-linear problems.
Some authors distinguish computer algebra from symbolic computation using the latter name to refer to kinds of symbolic computation other than the computation with mathematical formulas. Some authors use symbolic computation for the computer science aspect of the subject and "computer algebra" for the mathematical aspect. In some languages the name of the field is not a direct translation of its English name. Typically, it is called calcul formel in French, which means "formal computation". This name reflects the ties this field has with formal methods.
Symbolic computation has also been referred to, in the past, as symbolic manipulation, algebraic manipulation, symbolic processing, symbolic mathematics, or symbolic algebra, but these terms, which also refer to non-computational manipulation, are no more in use for referring to computer algebra.
There is no learned society that is specific to computer algebra, but this function is assumed by the special interest group of the Association for Computing Machinery named SIGSAM (Special Interest Group on Symbolic and Algebraic Manipulation).