Good morning,
Let F a field with valuation and $O_F$ its valuation ring. I have to consider $PGLn(F)/PGLn(O_F)$. I understood that PGLn(F) is the quotient of GLn(F) by the scalar diagonal matrix. Is then $PGLn(O_F)$ the quotient of $GLn(O_F)$ by the scalar diagonal matrices in $GLn(O_F)$ ? Then I don't get how do the quotient $PGLn(F)/PGLn(O_F)$ since $PGLn(O_F)$ is not a subgroup of $PGLn(F)$, if I understood correctly.
The definition of $\mathrm{PGL}_n(R)$ is subtle for general rings, but for local rings it's what one expects: it's $\mathrm{GL}_n(R)/R^\times$ (where $R^\times$ is embedded diagonally). If you're curious, there's always an injection $\mathrm{GL}_n(R)/R^\times\hookrightarrow \mathrm{PGL}_n(R)$ and whose cokernel can be identified with a subset of the Picard group $\mathrm{Pic}(R)$ (so, in particular, is trivial if $R$ is local!).
The set $\mathrm{PGL}_n(F)/\mathrm{PGL}_n(\mathcal{O}_F)$ is, as written on the tin, just a set. To understand this set, the key phrase is 'Iwasawa decomposition'. In particular, there is a decomposition $\mathrm{PGL}_n(F)=B(F)\mathrm{PGL}_n(\mathcal{O}_F)$ where $B$ is the (standard) Borel subgroup of $\mathrm{PGL}_n$ consisting of upper triangular matrices up to scalar. This should allow you to understand the set more concretely, I hope.
Hope this helps!