$H^1(G,A)$ with $G$ a Galois group and possibly noncommutative $A$ shows up in the theory of Galois descent. There are many "incarnations" of this, but here the general idea. Let $X$ be some kind of "object defined over a field $K$" (it might be a projective variety, an algebraic group, a finite-dimensional vector space, a finite-dimensional algebra etc.). Let $L/K$ be a Galois extension. Then say that a form of $X$ with respect to the extension $L/K$ is another object $Y$ defined over $K$ of the same kind such that the base changes $X_L \cong Y_L$ are isomorphic over $L$. In a lot of situations, the isomorphism classes of forms are classified by the Galois cohomology group $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}(X_L)$. A proof for a general statement can be found in Serre's book on Galois cohomology.

Some examples: let's work in the category of vector spaces and consider the vector space $X=K^n$. In this case, we can use that vector spaces are isomorphic iff they have the same dimension to see that $X_L\cong Y_L\implies X \cong Y$. Via the above statement, this implies that we get a generalized Hilbert 90 which says that $H^1(\mathrm{Gal}(L/K),\operatorname{GL}_n(L))$ vanishes.

Here's a slight variation: one can consider a group $H$, unrelated to the Galois group and fix a representation of $H$ on a finite-diimensional $K$-vector space $V$. Then we can also consider the set of $K$-representations $W$ such that $V_L \cong W_L$. Again these are classified $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}_{L[H]}(V_L))$ and again this vanishes, but this is a more nontrivial result, it follows from the Noether-Deuring theorem. In case $V$ is semisimple (using e.g. Maschke), you can also easily prove it by using Artin-Wedderburn and the generalized Hilbert 90 above.

Now let's work in the category of varieties and consider $X=\Bbb P_K^n$, the projective space. Then we get that $\mathrm{Aut}(X_L)\cong \mathrm{PGL}_{n+1}(L)$. Forms of $X$ are called Severi-Brauer varieties and they are classified by $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L))$. They are related to the Brauer group: we get from the exact sequence $1 \to L^\times \to \operatorname{GL}_{n+1}(L) \to \mathrm{PGL}_{n+1}(L) \to 1$. A morphism $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L)) \to H^2(\mathrm{Gal}(L/K),L^\times)$ which is injective by Hilbert 90. One can show that every element of the Brauer group lies in the image of this for some $n$.

One can also take other projective varieties (though there's not necessarily a relation to the Brauer group as above).

Here's a more exotic example: consider the standard octonion algebra $\mathbb O_K$ over $K$ (same structure constants as over $\Bbb R$). Then forms of this are general octonion algebras over $K$. The automorphism group $\mathrm{Aut}(\mathbb O_L)$ is the set of $L$-rational points of the exceptional algebraic group $G_2$. We get that $H^1(\mathrm{Gal}(K^{sep}/K),G_2(K^{sep}))$ classifies generalized octonion algebras.

One can also take other finite-dimensional algebras over $K$.

$H^1(G,A)$ with $G$ a Galois group and possibly noncommutative $A$ shows up in the theory of Galois descent. There are many "incarnations" of this, but here the general idea. Let $X$ be some kind of "object defined over a field $K$" (it might be a projective variety, an algebraic group, a finite-dimensional vector space, a finite-dimensional algebra etc.). Let $L/K$ be a Galois extension. Then say that a form of $X$ with respect to the extension $L/K$ is another object $Y$ defined over $K$ of the same kind such that the base changes $X_L \cong Y_L$ are isomorphic over $L$. In a lot of situations, the isomorphism classes of forms are classified by the Galois cohomology group $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}(X_L)$. A proof for a general statement can be found in Serre's book on Galois cohomology.

Some examples: let's work in the category of vector spaces and consider the vector space $X=K^n$. In this case, we can use that vector spaces are isomorphic iff they have the same dimension to see that $X_L\cong Y_L\implies X \cong Y$. Via the above statement, this implies that we get a generalized Hilbert 90 which says that $H^1(\mathrm{Gal}(L/K),\operatorname{GL}_n(L))$ vanishes.

Here's a slight variation: one can consider a group $H$, unrelated to the Galois group and fix a representation of $H$ on a finite-diimensional $K$-vector space $V$. Then we can also consider the set of $K$-representations $W$ such that $V_L \cong W_L$. Again these are classified $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}_{L[H]}(V_L))$ and again this vanishes, but this is a more nontrivial result, it follows from the Noether-Deuring theorem.

Now let's work in the category of varieties and consider $X=\Bbb P_K^n$, the projective space. Then we get that $\mathrm{Aut}(X_L)\cong \mathrm{PGL}_{n+1}(L)$. Forms of $X$ are called Severi-Brauer varieties and they are classified by $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L))$. They are related to the Brauer group: we get from the exact sequence $1 \to L^\times \to \operatorname{GL}_{n+1}(L) \to \mathrm{PGL}_{n+1}(L) \to 1$. A morphism $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L)) \to H^2(\mathrm{Gal}(L/K),L^\times)$ which is injective by Hilbert 90. One can show that every element of the Brauer group lies in the image of this for some $n$.

One can also take other projective varieties (though there's not necessarily a relation to the Brauer group as above).

Here's a more exotic example: consider the standard octonion algebra $\mathbb O_K$ over $K$ (same structure constants as over $\Bbb R$). Then forms of this are general octonion algebras over $K$. The automorphism group $\mathrm{Aut}(\mathbb O_L)$ is the set of $L$-rational points of the exceptional algebraic group $G_2$. We get that $H^1(\mathrm{Gal}(K^{sep}/K),G_2(K^{sep}))$ classifies generalized octonion algebras.

One can also take other finite-dimensional algebras over $K$.

$H^1(G,A)$ with $G$ a Galois group and possibly noncommutative $A$ shows up in the theory of Galois descent. There are many "incarnations" of this, but here the general idea. Let $X$ be some kind of "object defined over a field $K$" (it might be a quasiprojective variety, an algebraic group, a finite-dimensional vector space, a finite-dimensional algebra etc.). Let $L/K$ be a Galois extension. Then say that a form of $X$ with respect to the extension $L/K$ is another object $Y$ defined over $K$ of the same kind such that the base changes $X_L \cong Y_L$ are isomorphic over $L$. In a lot of situations, the isomorphism classes of forms are classified by the Galois cohomology group $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}(X_L)$. A proof for a general statement can be found in Serre's book on Galois cohomology.

Some examples: let's work in the category of vector spaces and consider the vector space $X=K^n$. In this case, we can use that vector spaces are isomorphic iff they have the same dimension to see that $X_L\cong Y_L\implies X \cong Y$. Via the above statement, this implies that we get a generalized Hilbert 90 which says that $H^1(\mathrm{Gal}(L/K),\operatorname{GL}_n(L))$ vanishes.

Here's a slight variation: one can consider a group $H$, unrelated to the Galois group and fix a representation of $H$ on a finite-diimensional $K$-vector space $V$. Then we can also consider the set of $K$-representations $W$ such that $V_L \cong W_L$. Again these are classified $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}_{L[H]}(V_L))$ and again this vanishes, but this is a more nontrivial result, it follows from the Noether-Deuring theorem. In case $V$ is semisimple (using e.g. Maschke), you can also easily prove it by using Artin-Wedderburn and the generalized Hilbert 90 above.

Now let's work in the category of varieties and consider $X=\Bbb P_K^n$, the projective space. Then we get that $\mathrm{Aut}(X_L)\cong \mathrm{PGL}_{n+1}(L)$. Forms of $X$ are called Severi-Brauer varieties and they are classified by $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L))$. They are related to the Brauer group: we get from the exact sequence $1 \to L^\times \to \operatorname{GL}_{n+1}(L) \to \mathrm{PGL}_{n+1}(L) \to 1$. A morphism $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L)) \to H^2(\mathrm{Gal}(L/K),L^\times)$ which is injective by Hilbert 90. One can show that every element of the Brauer group lies in the image of this for some $n$.

One can also take other projective varieties (though there's not necessarily a relation to the Brauer group as above).

Here's a more exotic example: consider the standard octonion algebra $\mathbb O_K$ over $K$ (same structure constants as over $\Bbb R$). Then forms of this are general octonion algebras over $K$. The automorphism group $\mathrm{Aut}(\mathbb O_L)$ is the set of $L$-rational points of the exceptional algebraic group $G_2$. We get that $H^1(\mathrm{Gal}(K^{sep}/K),G_2(K^{sep})$ classifies generalized octonion algebras.

One can also take other finite-dimensional algebras over $K$.

$H^1(G,A)$ with $G$ a Galois group and possibly noncommutative $A$ shows up in the theory of Galois descent. There are many "incarnations" of this, but here the general idea. Let $X$ be some kind of "object defined over a field $K$" (it might be a projective variety, an algebraic group, a finite-dimensional vector space, a finite-dimensional algebra etc.). Let $L/K$ be a Galois extension. Then say that a form of $X$ with respect to the extension $L/K$ is another object $Y$ defined over $K$ of the same kind such that the base changes $X_L \cong Y_L$ are isomorphic over $L$. In a lot of situations, the isomorphism classes of forms are classified by the Galois cohomology group $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}(X_L)$. A proof for a general statement can be found in Serre's book on Galois cohomology.

Some examples: let's work in the category of vector spaces and consider the vector space $X=K^n$. In this case, we can use that vector spaces are isomorphic iff they have the same dimension to see that $X_L\cong Y_L\implies X \cong Y$. Via the above statement, this implies that we get a generalized Hilbert 90 which says that $H^1(\mathrm{Gal}(L/K),\operatorname{GL}_n(L))$ vanishes.

Here's a slight variation: one can consider a group $H$, unrelated to the Galois group and fix a representation of $H$ on a finite-diimensional $K$-vector space $V$. Then we can also consider the set of $K$-representations $W$ such that $V_L \cong W_L$. Again these are classified $H^1(\mathrm{Gal}(L/K),\operatorname{Aut}_{L[H]}(V_L))$ and again this vanishes, but this is a more nontrivial result, it follows from the Noether-Deuring theorem. In case $V$ is semisimple (using e.g. Maschke), you can also easily prove it by using Artin-Wedderburn and the generalized Hilbert 90 above.

Now let's work in the category of varieties and consider $X=\Bbb P_K^n$, the projective space. Then we get that $\mathrm{Aut}(X_L)\cong \mathrm{PGL}_{n+1}(L)$. Forms of $X$ are called Severi-Brauer varieties and they are classified by $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L))$. They are related to the Brauer group: we get from the exact sequence $1 \to L^\times \to \operatorname{GL}_{n+1}(L) \to \mathrm{PGL}_{n+1}(L) \to 1$. A morphism $H^1(\mathrm{Gal}(L/K),\mathrm{PGL}_{n+1}(L)) \to H^2(\mathrm{Gal}(L/K),L^\times)$ which is injective by Hilbert 90. One can show that every element of the Brauer group lies in the image of this for some $n$.

One can also take other projective varieties (though there's not necessarily a relation to the Brauer group as above).

Here's a more exotic example: consider the standard octonion algebra $\mathbb O_K$ over $K$ (same structure constants as over $\Bbb R$). Then forms of this are general octonion algebras over $K$. The automorphism group $\mathrm{Aut}(\mathbb O_L)$ is the set of $L$-rational points of the exceptional algebraic group $G_2$. We get that $H^1(\mathrm{Gal}(K^{sep}/K),G_2(K^{sep}))$ classifies generalized octonion algebras.

One can also take other finite-dimensional algebras over $K$.