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Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.

Is this only for $n=2$?

Is there any relation between $H^1(G_K,PGL_n)$ and the Brauer group for $n>2$?

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    $\begingroup$ This is discussed, for instance, in Serre's book "Galois Cohomology". Assuming that the characteristic is prime to $n$, there is a map $\Delta:H^1(G_K,\text{PGL}_n) \to H^2(G_K,\mu_n)$. The theory of central simple algebras shows that this map is injective. It is frequently not surjective. However, it is surjective for many classes of fields of cohomological dimension $2$. Such results are frequently called "Period-Index Theorems", e.g., the Period-Index Theorem of Johan de Jong. $\endgroup$
    – Jason Starr
    Oct 20, 2016 at 14:08

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Your first statement is not quite true. A $\mathrm{PGL}_2$-torsor does indeed give an element of order $2$ in the Brauer group, but there can be elements of order $2$ in the Brauer group of a general field $K$ that are not represented by quaternion algebras, so do not come from $\mathrm{PGL}_2$-torsors. What is true is that the image generates the $2$-torsion in the Brauer group: this follows from the Merkurjev-Suslin theorem. Part of the reason this is confusing is that $\mathrm{H}^1(K,\mathrm{PGL}_n)$ is not a group, since $\mathrm{PGL}_n$ is not commutative. Look up the period-index problem for more details.

That said, there is indeed a natural map $\mathrm{H}^1(K,\mathrm{PGL}_n) \to \mathrm{Br}(K)$ for arbitrary $n$: it comes from the exact sequence in (non-Abelian) cohomology coming from the short exact sequence

$0 \to \mathbb{G}_m \to \mathrm{GL}_n \to \mathrm{PGL}_n \to 0$

of algebraic groups over $K$, together with the identification $\mathrm{Br}(K) = \mathrm{H}^2(K,\mathbb{G}_m)$. To see that the image lands in the $n$-torsion, compare this with the corresponding sequence for $\mathrm{SL}_n$ and use the Kummer sequence.

There are lots of references for this kind of stuff; an excellent one is Central simple algebras and Galois cohomology by Gille and Szamuely.

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