If I tell you that Corinthia is my second cousin, once removed, there are two major problems:
- You probably don't know what the heck that means
- If you do know, you still don't know whether she is one generation older, or one generation younger.
This nomenclature has been around for a long time, and it's just not working. So we need a new system. From now on, we'll all use the system outlined below.
Relatives (or even close friends from your childhood), who are in the generation above you are aunts and uncles; the generation above that are great-aunts and great-uncles; and so on (great-great is awkward, but really, how often do you need it?).  This applies to all your relatives: if you know the relationship well enough to know that they are the generation above you, then they are your aunt (or uncle) and you call them "Aunt" (or "Uncle").
Relatives and close-enough friends in your generation are siblings (sometimes, brothers or sisters). We'll get to "cousins" below.
The generation below you are your nieces and nephews. Then great-nieces and great-nephews. And so on.
Obviously, we will sometimes need to be able to clarify which of your aunts and uncles (and so on) are what we would now call "real" (of course, they're all real, but you know what I mean). Thus, we will optionally use the following clarifiers:
Genealogists, or even just people who are really into all this, will also of course sometimes divide Distant into Third, Fourth, etc.
Cousins don't really fit into our system, but we're obviously not going to get people to drop the word any time soon (at least for people in their own generation). We keep the original idea of cousins by simply saying that you can call a second sibling a first cousin, a third sibling a second cousin, and so on. Starting now, it is always fine to call your cousin a sibling, or brother or sister, or a second brother, or whatever.
The number attached to the relationship is the number of generations down from the common ancestor to the older person in the relationship: if your grandmother and my father were siblings, that means that their parents (my grandparents) are our common ancestors. That's two generations from me (the older one), so I am your second uncle.
You can also count from a known starting point.
- If I already know that your parents are my second siblings (first cousins), I can move down a generation. Since I am not moving the older generation away from the common ancestor, I am still your second uncle.
- If you already know that my mother is your first great-aunt, you can move down one generation. Since you are moving the older person (from my mother to me), you increase one "degree" when you move down one generation, and I am still your second uncle.
Alice: That's my Aunt Terry Bob: Is she your first aunt? Alice: No, she's a friend Aunt.
Alice: That's my Uncle Terry. Bob: Is he your first uncle? Alice: No, he's my Mom's first cousin, so he's my second uncle.
Alice: That's my Cousin Terry. Bob: Cut it out.
We have special words for people in the direct line of descent:
- Your zero-th great-aunt (or -uncle) is your "grandparent"
- Your zero-th aunt (or uncle) is your "parent"
- Your zero-th sibling is your "self"
- Your zero-th niece (or nephew) is your "child"
- ... and so on.
English does not distinguish between aunts, uncles, nephews and nieces who are connected directly, vs. those who are connected by marriage, in the older generation.
This will be generalized as follows:
- We will all agree that since this rule applies to the older generation, it should also apply to the younger generation: thus, your niece's husband is your nephew.
- We will all agree that since it applies to nieces, nephews, aunts and uncles, it also applies to siblings (and cousins).
- We will all agree that since this rule applies to marriage, it applies to other stable partnerships, as defined by the partners. Thus the man your nephew has been living with for the last 20 years is your nephew (if he wants to be). 
Optionally, clarify by using
- Direct (no partnership in between)
- Indirect (one or more partnerships, this can be further qualified by single or double)
Alice: Cousin Terry is my double indirect first cousin. Bob: If you say so. Alice: His partner Pat is my partner Pat's direct first cousin. Bob: Seriously?
Step- and half-
Step- and half- can continue to be used, when appropriate, to clarify changing partnership and parental relationships, just as they are now. They are strictly optional.
We're all going to do it this way from now on, or I'll hold my breath until I turn blue.
- From now on, "great" and "grand" can be used interchangeably, in various combinations: "great nephew" and "grand nephew", or "great-great nephew" and "great-grand nephew" are all fine. You're welcome.
- Remember, cousins are special. Use the sibling number when calculating,and convert to cousins later if you want to.
- Meaning, it is the people in the partnership who get to define whether the partnership counts for this purpose. People who get married, legally or otherwise, are declaring publicly that they want it to.
Corinthia is actually my third aunt. Now you know.