This version requires PHP 5.3.4 or later, due to show-stopper bugs in earlier versions of PHP 5.3. Important note on reading PHP versions: PHP 5.3.20 is newer than 5.3.10 which is newer than 5.3.4. Moreover, PHP 5.4.0 is newer than PHP 5.3.4. Joomla! 3.3 and later already requires PHP 5.3.10 or later. Also keep in mind that the entire PHP 5.3 series is end of life since August 2014 and MUST NOT be used on live sites any more for performance and security reasons.
This release requires any version in the Joomla! 3.3.x or 3.4.x family. It is NOT compatible with older versions of Joomla! including all 1.x, 2.x, 3.0.x, 3.1.x and 3.2.x releases. In any case, if your server doesn't meet the minimum requirements our pre-installation check asks Joomla! to cancel the installation and tell you exactly why it cannot be installed (minimum PHP or Joomla! version not satisfied).
Reading Joomla! versions: Joomla! 2.5.20 is newer than 2.5.10 which is newer than 2.5.6. Moreover, Joomla! 3.4.0 is newer than 3.3.6 which is newer than 2.5.28. In other words, you need to separately compare each one of the the three numbers separated by dots. These numbers are NOT decimals and trailing zeroes DO matter, i.e. 1.10 (pronounced "one dot ten", not one dot one oh) is larger than 1.2.