phpSATk requires PHP 5.*. phpSATk is developed on bleeding edge PHP Versions (5.2.2 at the moment), there may be (and are) issues when using certain older version. For security reasons I advice you to use Suhosin patched Versions of PHP.
phpSATk is known to run on Windows and Linux, but will probably run on other Unix derivates.
Webservers supported are Apache and LigHTTPd.
The core itself requires no external PHP modules (DOM is required but enabled by default), optionally you might want to install Pecl-Memcache and use memcached caching (which is faster).
Big parts of the output system rely on PHPTAL, make sure you have installed it (into your include path).
Other modules may have external depencies:
Some core subsystems (most noticably authentication/authorization) require a database, for which I use MySQL (4.1+). Other RDBMS might work but I don’t know.