The Balrogs (Sindarin for ‘Demon of Might’; Quenya Valarauko) were Maiar (“Ainur of lesser degree than the Valar”). Other Maiar were Gandalf, Sauron, Melian. Gothmog and “Durin’s Bane” are the only two named.
Balrogs would be under the Maiar category, but most have no recorded name. I suppose the one from the trilogy could have been listed as "Durin's Bane."
The Balrogs were Maiar, a category that includes the wizards (Gandalf, Saruman, etc.) and Sauron. The Balrog Gothmog is there, but not the one that appears in the Fellowship of the Ring, possibly because it isn't given a name.
Basically bad Maiar.