The magic_quotes_gpc feature automatically escapes single quotes to make it possible to store them in a database. Sql queries use single quotes to define values, like this: update product set name='New Name'. When a text string contains unescaped single quotes it can't be used in an sql query.
Currently AShop won't work with magic_quotes_gpc turned off. If it is off, ask your hosting provider to turn magic_quotes_gpc on. It is just a matter of changing the word "off" to "on" in php.ini and restarting the web server. It is not a security risk and most hosting companies use this setting.
The settings for magic_quotes to work with AShop Deluxe are:
magic_quotes_gpc = On
magic_quotes_runtime = Off
magic_quotes_sybase = Off