A finitely presented algebra (fp-algebra) in Magma is simply the quotient ring of a free algebra F = R< x1, ..., xn > by an ideal J of F. It is an object of type AlgFP with elements of type AlgFPElt.
The elements of fp-algebras are simply noncommutative polynomials which are always kept reduced to normal form modulo the ideal J of "relations". Practically all operations which are applicable to noncommutative polynomials are also applicable in Magma to elements of fp-algebras (when meaningful).
If an fp-algebra A has finite dimension, considered as a vector space over its coefficient field, then extra special operations are available for A and its elements.