See @MikeSynnott's answer for the correct technical terms.
As he notes, you're looking for familiar terms, but it's not easy to give you an answer that will satisfy you.
Once you get beyond the immediate relationships like "uncle" that are clear and easy to define, it often comes down to family preference as to how people refer to each other, and usage may differ from family to family, or even within a family.
For example, my Uncle Joe is my father's brother, so is my uncle. However, my children also refer to him as Uncle Joe, even though technically he is their Great Uncle. But my mother's sister, Betty is referred to by all the family as Great Aunt Betty; even my siblings who don't have children call her that, despite her not having a 'Great Aunt' relationship to them at all.
Why do we do that? I don't honestly know. It's just how the family works. Joe is a warm, fatherly figure with a close relationship to us all, and calling him "Uncle" helps to cement that. Betty is more of a matriarchal, older-generation figure; calling her "Great Aunt" seems to add dignity and gravitas.
Another example is my cousin's daughter. Technically she is my "first cousin, once removed". However, we just refer to each other as cousins, because it's easier that way. My kids also just call her "cousin", even though technically for them he is their "second cousin". Same with my half-sister; we never bother with the prefix "half" because it diminishes the relationship. We're just brother and sister.
Overall, I think in most cases, where you have a family relationship that is close enough that people consider themselves part of the same family, they will generally just use the single-word "cousin", "aunt" or "uncle"; whichever is most appropriate. The "great" qualifier may be used in some cases, but probably not much else. People will only likely use the actual technically correct terms when they actually need to be technically correct about the relationship; for example, when they're explaining the family relationship to an outsider. But within the family, they generally won't bother.
The only other term that I've heard used which may be useful is "distant cousin" (or "distant uncle", etc), which may be used when describing a family relationship that is further removed than the simple direct cousin relationship, but where you don't feel the need to discuss the exact relationship (or where you simply don't know it). Examples: "A distant uncle died, and left me a small inheritance". Or "Hi. You don't know me, but we're distant cousins, and I'm organising a family re-union."