Binding bigraphs are a graphical formalism intended to be a meta-model for mobile, concurrent and communicating systems. In this paper we present an algebra of typed graph terms which correspond precisely to binding bigraphs over a given signature. As particular cases, pure bigraphs and local bigraphs are described by two sublanguages which can be given a simple syntactic characterization. Moreover, we give a formal connection between these languages and Synchronized Hyperedge Replacement algebras and the hierarchical graphs used in Architectural Design Rewriting. This allows to transfer results and constructions among formalisms which have been developed independently, e.g., the systematic definition of congruent bisimulations for SHR graphs via the IPO construction.