The Debian website has an extensive set of translations of the pages.
But when one language is selected, that choice is not followed through
when clicking on links.

For instance clicking on the Italiano link on the home page
(https://www.debian.org/) brings me to the italian translation of the
home page, https://www.debian.org/index.it.html

But then if I click on, say, "La nostra filosofia" (i.e. "Our
Philosophy"), the Italian page brings me to
https://www.debian.org/intro/philosophy, which is back to English. The
link on the Italian home page should bring me to the Italian version
of the Philosophy page (it does exist,
https://www.debian.org/intro/philosophy.it.html)

The problem is general with the Debian website, it affects other
languages and other links.

What language the website is shown in is dependant on what language
your browser is setup to show. (It's a browser setting).

As you have noticed, the links at the bottom of each page does only
change to the translated version of the current page, if you then click
another link it will go back to the browser set language again.

To make the complete Debian website show in another language, change
the language setting in your browser. How this is done is described in
detail on

https://www.debian.org/intro/cn

If you for some reason don't want to change the browser language
setting, there is a way to set a cookie in the browser that also will
change the shown language. This too is described in the page I just linked.

That approach seems suboptimal, and not consistent with the behaviour
usually experienced on other websites.

For instance, the Wikipedia article on Debian is
https://en.wikipedia.org/wiki/Debian .  If I select language to Italian,
it sends me to https://it.wikipedia.org/wiki/Debian . If I select links
from there, e.g. "software libero" ("free and open-source software"), it
remains in italiano, e.g. forwarding to
https://it.wikipedia.org/wiki/Software_libero .  No fiddling about with
system language or cookies required.

Another example, the KTH university site. https://www.kth.se/ goes to
the Swedish home page. If I click on "International website", it goes to
https://www.kth.se/en in English.  Then if I click on "About KTH", for
instance, it forwards to https://www.kth.se/en/om, still in English. It
doesn't revert back to the original Swedish page, https://www.kth.se/om.

Sometimes you just want to inspect the information in the other
language, e.g. to show to a friend who speaks that language. It doesn't
always mean you want the default langauge on your system to be set to
that language.