To add to George's answer above, 1st Galois cohomology group has a natural interpretation as the set of classes of principal homogeneous spaces for a group.
Let $G$ be an algebraic group defined over $k$. Let $K/k$ be a Galois extension. The $K$-points of $G$ form a $\mathrm{Gal}(K/k)$-group (or a $\mathrm{Gal}(K/k)$-module, if $G$ is commutative). A principal homogeneous $K$-space is a $K$-variety $X$ with a free transitive action of $G$. There is a bijective correspondence between $k$-isomorphism classes of principal homogeneous $K$-spaces and cocycles of $H^1(\mathrm{Gal}(K/k), G(K))$.
Let $X$ be a p.h.s. Pick a point $x \in X$ and act on it by $\sigma \in \mathrm{Gal}(K/k)$. Since $G$ acts on $X$ freely and transitively there is a unique $g_\sigma$ such that $g_\sigma \cdot x = \sigma(x)$. One checks that $\sigma \mapsto g_\sigma$ define a 1-cocycle.
Conversely, given a 1-cocyle $\{g_\sigma\}$, consider a disjoint union $\sqcup G \times \{\sigma\}_{\sigma \in \mathrm{Gal}(K/k)}$ and define an action of $\mathrm{Gal}(K/k)$ on it: $\sigma(x,\tau)=(g_\sigma \cdot x, \sigma\tau)$. There is a natural $G$ action on the disjoint union, and the factor by the action $\mathrm{Gal}(K/k)$ inherits the action of $G$ (a highbrow term for what is happening is "Galois descent"), turning it into a p.h.s.
Moreover, if $G$ is commutative, the group law on $H^1(\mathrm{Gal}(K/k), G(K))$ has a natural geometric interpretation in terms of principal homogeneous spaces. This group is also known as Weil-Chatelet group. I recommend the original Weil's article on the subject: A. Weil, "On algebraic groups and homogeneous spaces". Amer. J. Math. , 77 (1955) pp. 493–512