There are three staple bitters needed to make the majority of classic cocktails, plus one “good to have”.
These are the quintessential aromatic bitters. They have been in existence since the early 19th century and are required for many cocktails including the Manhattan and the Old Fashioned. They are not actually bittered with angostura bark, but named for the region of their creation.
Orange bitters are not a brand, but a style. Many brands exist today, however I use Angostura orange bitters. These bitters are an integral component of the Martini.
Another brand of aromatic bitters, these are required for the Sazerac cocktail.
These are the least relevant bitters needed when starting out. However, they are called for in many 19th century cocktails and are worth buying if you want to make a classic Martinez cocktail. The original brand is no longer in production, but certain recreations including Dr. Adam Elmegirab’s Boker’s Bitters and The Bitter Truth’s Bogart’s Bitters.