Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that x·g = y, where · denotes the (right) action of G on X).
To state the definition more explicitly, X is a G-torsor or G-principal homogeneous space if X is nonempty and is equipped with a map (in the appropriate category) X × G → X such that for all x ∈ X and all g,h ∈ G, and such that the map X × G → X × X given by is an isomorphism (of sets, or topological spaces or ..., as appropriate, i.e. in the category in question).
Note that this means that X and G are isomorphic (in the category in question; not as groups: see the following).
(This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.)
One way to follow basis-dependence in a linear algebra argument is to track variables x in X.
A global section will exist (by definition) only when M is parallelizable, which implies strong topological restrictions.
In number theory there is a (superficially different) reason to consider principal homogeneous spaces, for elliptic curves E defined over a field K (and more general abelian varieties).
The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed.
That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K).
The curves C turn out to be torsors over E, and form a set carrying a rich structure in the case that K is a number field (the theory of the Selmer group).
In this case, a (right, say) G-torsor E on X is a space E (of the same type) over X with a (right) G action such that the morphism given by is an isomorphism in the appropriate category, and such that E is locally trivial on X, in that E → X acquires a section locally on X. Isomorphism classes of torsors in this sense correspond to classes in the cohomology group H1(X,G).
When we are in the smooth manifold category, then a G-torsor (for G a Lie group) is then precisely a principal G-bundle as defined above.