I use Debian 6 and successfully installed Nginx as webserver (I can see my html test file).

Now I would like to install PHP with some mods I need. I tried:

$ sudo apt-get install php5 php5-fpm php5-mysql php5-common php5-curl php5-cli

But it results in

I already tried to

$ sudo apt-get upgrade

On my Raspberry Pi with Raspbian (= Debian optimized for the RPi) the necessary php5-fpm package is in its repository.

How can I install php5-fpm manually (without using a LEMP stack)?
Do I need to add some extra repository for it? Which & how?

It looks like you need to add the Dotdeb repository.

Great! Thanks to your answer I found http://www.dotdeb.org/instructions/ and was able to add the right repository & install php5-fpm.
Thanks very much!