Open_Basedir Restriction

This error means that a script is trying to run (usually the SQL install script for an extension) in a directory where scripts are not allowed to run as defined in your php.ini or Apache config file (depending on your configuration). This is a security feature designed to disallow untrusted scripts to run - more info here
To resolve this you should either comment out the line or add in the directory where the script lies, on a standard install the line is commented out (see screen shot attached).
The full error will look something like this:
a:5:{i:0;s:509:"Error in file: "/var/www/vhosts/your-domain/httpdocs/app/code/local/Auctionmaid/Productmatrix/sql/productmatrix_setup/mysql4-install-3.0.2.php" - Warning: realpath() [<a href='function.realpath'>function.realpath</a>]: open_basedir restriction in effect. File(your install path) is not within the allowed path(s): (/var/www/vhosts/your domain/httpdocs:/tmp)  in /var/www/vhosts/your domain/httpdocs/app/code/local/Auctionmaid/Productmatrix/sql/productmatrix_setup/mysql4-install-3.0.2.php on line 2";i:1;s:1060:"#0 /var/www/vhosts/your-domain/httpdocs/app/code/core/Mage/Core/Model/Resource/Setup.php(374): Mage::exception('Mage_Core', 'Error in file: ...')