I remember finding a very useful algebraic structure that I loosely call "group over 2 sets" because I can't remember its real name or find it. I looked at all the algebraic structures on Wikipedia, but it wasn't mentioned there even though I remember it having its own page.
Anyway, I'll define the algebraic structure more formally:
It has 2 sets $P$ and $D$, where $D$ is a group over addition, 2 operations $+ : P \times D \rightarrow P$ and $- : P \times P \rightarrow D$ and few laws:
$$ x + (y - x) = y \\ (y - x) + (x - y) = 0 \\ (x - x) = 0 \\ x + 0 = x $$
And possibly some more laws. I'm not sure I remember them all.
I remember the structure being used for describing the relation between e.g. points and vectors (the set of points being P and vectors being D) or times = P and timedifferences = D.
Does anyone know how this algebraic structure is called?
This is called a torsor: more precisely, it is the structure of a (right) $D$-torsor on the set $P$. There are various ways to axiomatize such a structure, but one simple way is to say that $D$ is a group which acts on the set $P$ (on the right) and for any $a,b\in P$, there is a unique $d\in D$ such that $ad=b$. In your notation, $+$ is the group action and $-$ is the map taking $(b,a)$ to $d$. (Of course, your additive notation is normally used only when the group $D$ is abelian.)
In the special case that $D$ is not just a group but a vector space, a $D$-torsor is also called an affine space, as rschwieb mentioned.