A torsor $T$ is just an algebra equipped with a ternary operation $a,b,c ∈ T ↦ abc ∈ T$ such that $abb = a = bba$ and $(abc)de = ab(cde)$.
The archetypical example of a torsor is a group $G$ whose product $a,b ∈ G ↦ a·b ∈ G$ and inverse $a ∈ G ↦ a^{-1} ∈ G$ yield the torsor operation $abc ≡ a·b^{-1}·c$. This operation is "affine" in the sense that it is covariant with respect to multiplication on left and right - $(g·a·h)(g·b·h)(g·c·h) = g·abc·h$; so the structure removes the "first class" standing of the group identity.
The group operations themselves can be recovered from the ternary operation, once a group identity e ∈ G has been specified, by defining $a·b ≡ aeb$, $a^{-1} ≡ eae$. Then it is a routine matter to show that the group axioms follow from the torsor axioms.
Relations suggested by this example, such as these
$$
(ade)(bde)(cde) = (abc)de,\quad (abc)(abd)(abe) = ab(cde),\\
(abe)(ace)(ade) = a(dcb)e,\quad (abc)de = a(dcb)e = ab(cde),
$$
can all be proven from the torsor axioms.
The torsors that are associated with Abelian groups are precisely those for which the identity $abc = cba$ also holds. Consequently, these may be termed Abelian torsors.
A torsor $T$ contains within it a natural group structure in two ways. First, each $e ∈ T$ can be taken as the identity of a group $T_e$ whose operations are defined as just indicated. This, you might call the "tangent group" at $e$. That they are all groups confirms the equal standing that all points have in the torsor: that any of them can be taken as the group identity.
Second, one can also define a formal quotient $a,b ∈ T ↦ a \backslash b ≡ ρ[(a,b)]$ with the equivalence classes taken with respect to the equivalence relation $ρ ∈ T×T$ generated from the relations $(cba,d) ρ (a,bcd)$. This implements the relation $cba \backslash d = a \backslash bcd$. The resulting algebra $δT ≡ (T×T)/ρ$ is a group with product $(a \backslash b)·(c \backslash d) = cba \backslash d = a \backslash bcd$ and (thus) a group identity equal to $a \backslash a$ for all $a ∈ T$, and an inverse $(a \backslash b)^{-1} = b \backslash a$.
This group acts on the torsor on the right by the action $a(b \backslash c) = abc$. You may confirm this by noting that
$$a(b \backslash c)(d \backslash e) = (abc)de = ab(cde) = a(b \backslash cde);$$
while the well-definedness of the operation follows since $a(dcb \backslash e) = a(dcb)e = ab(cde)$.
The tangent groups $T_e$, for each $e ∈ T$ are isomorphic to each other and to $δT$; with the isomorphism given by the correspondences
$$π_e: a ∈ T_e ↦ e \backslash a ∈ δT,\quad π_e^{-1}: a \backslash b ∈ δT ↦ eab ∈ T_e,$$
which are inverses of one another. The composition
$${π_f}^{-1} ∘ π_e: a ∈ T_e ↦ fea ∈ T_f$$
yields the isomorphisms between the respective tangent groups.
From this, you can also talk about a Lie torsor $T$ as a manifold whose torsor operation satisfies suitable smoothness properties. The tangent groups and group action then provide the structure of a principal bundle on the torsor $T$ with fibre $δT$. In fact, the very definition of principal bundles (and associated bundles) can themselves be rendered in a more transparent and direct fashion in an analogous way to what has been done here.
Also related are affine geometries, which bear the same relation to vector spaces as torsors do to groups. In addition to the Abelian torsor operation
$$a,b,c ∈ A ↦ abc = a - b + c ∈ A$$
of an affine geometry $A$, you also have the "barycenter" operation
$$a,c ∈ A, λ ∈ F ↦ [a,λ,c] = (1-λ)a + λc ∈ A,$$
where $F$ is the underlying field.
To recover the torsor properties, and to recover the structure of a vector space over the field $F$ for the fibres $A_o$, for each $o ∈ A$, it is enough to postulate the following identities
$$[a,0,c] = a,\quad [a,1,c] = c,\quad [a,λν(1-ν),[b,μ,c]] = [[a,λν(1-μ),b],ν,[a,λμ(1-ν),c]];$$
and to define the torsor operation by
$$abc = [b,1/(1-λ),a],λ,[b,1/λ,c]],\quad (λ ∈ F - {0,1})$$
... the independence from $λ$ following as a consequence of the other properties. This suffices to characterize Affine geometries over all fields except the 2 and 3 element fields; which must be handled specially.
For the 2-element field, the barycenter operation is trivial, and the torsor operation has the additional identity $a - b + a = b$. For the 3-element field, the barycenter operation is reducible to the torsor operation, with
$$[a,0,b] = a,\quad [a,+1,b] = b,\quad [a,-1,b] = a - b + a = b - a + b = [b,-1,a].$$
You may want to try the exercise of drawing the correspondences to other definitions for torsors found in the literature, just to see how they relate to this and to make the ideas described by them accessible - and more transparent.
