There is only one species of budgerigar, so any different names you might meet in print such as ‘English’, ‘American’ or ‘Australian’ is just a detail - all of them are Melopsittacus undulates. That being said, these three broad groupings are commonly used when writing about the birds, especially in relation to bird shows and exhibitions.
‘Australian’ refers to birds confined to Australia (following the export ban of 1894), these birds generally resemble their wild ancestors in terms of size. ‘American’ pet parakeets are slightly bigger than Austrralian budgies, but are still close to the original wild birds in terms of shape. ‘English’ budgies are noticeably larger, and have more variation in feather length and thickness.
The budgerigar or parakeet - one species, but lots of varieties
All three of these broad groupings of parakeets come in a handful of varieties and a range of color types, rather than separate breeds. Little interbreeding between the types takes place, and the gene pool is relatively limited, meaning that even when two birds do breed, there are only a few variations in size and feathers. So whether your parakeet is larger than average, sports a tuft of feathers on its head, or has a striking all-yellow plumage, it will still be unmistakably a parakeet in shape, movement, voice, and behavior.