Since you asked....
Biznaga (also sometimes spelled bisnaga, viznaga, visnaga) is the Spanish word for a round, spiny cactus. Like many other common names it doesn't always have a very specific meaning. There is also a genus of non-succulents called Visnaga, just to confuse things.
The golden barrel would be a biznaga, as would other barrel cacti. In Mexico the term often refers to a specific class of plants which have been used to prepare a sort of candy called acitrón, which is traditionally consumed on the Dia de Reyes (Jan. 6) as part of the rosca de reyes (a cake/bread associated with that day). I don't know how much the common name is used outside of Mexico but it is said to apply to some South American cacti as well. As a side note, acitrón is bad stuff unless you can prove it came from a cultivated cactus (and good luck with that). There has been a movement (somewhat but not very successful) to reduce the mass harvest of big old wild cacti for the sake of a snack that is mostly introduced sugar anyway.
One species of biznaga particularly associated with acitrón in Mexico is Echinocactus platyacanthus, and if you had to assign that common name to one plant only, that's the one I would choose. Next in line would be Ferocactus histrix, which is very similar looking.
Beyond the big stout spines of Ferocactus and Echinocactus, various other genera have been associated with the name biznaga. I would not personally call any Mammillaria a biznaga, but I'm no kind of authority on this sort of thing. If you do an internet search for Mammillaria and biznaga you will come across any number of common names like "biznaga de chilitos" (which most Mammillarias are, as their fruit is chile-like in shape and color) and "biznaga espinosa" (spiny biznaga), but in all fairness just about all Mammillarias are spiny. That particular combination seems redundant to me. So...
As well as being a common name, I think Biznaga is also a subgenus or section of Ferocactus, and this relates to the type of fruit it bears. But the details escape me at this point, and that's a level of taxonomic detail that I don't bother remembering.
