simpleContact's requirements are deliberately modest. You need a web server with PHP support and access to a MySQL database.
MySQL 4.1 is strongly recommended because of its support for varying character sets and collations (which aids internationalisation). Although earlier versions may work, version 4.0 is the oldest version that I actively support.
A definitive minimum spec for PHP is a work-in-progress. PHP 4 and above are recommended.
The built-in update check requires either fopen or cURL.
The "remember me" checkbox at login uses a cookie.