I am making a web page for a company and they need to add and remove listings every week. How can i do this / is it possible?
I Think: Yes it's possible but it's done via PHP coding, and databases (MYSQL). From my limited experience you would have to have knowledge of PHP/MYSQL. It depends on how big the task is, for example when they login how many options do they have? If you need a high-powered GUI then you could install a readily available CMS like Wordpress for free, or pay someone to do it for you. Note the "I Think" because i'm not 100% certain but just to give you the basic idea.
You can search google for a login script tutorial and that should not be too hard. But the deleting listings will be slightly more hard. You will need to use PHP and MySQL like wd_2k6 said. You can make the structure of the SQL database however you need it then using PHP you can tell it to delete whatever entries you need it to. Just use google, there are thousands of PHP tutorials out there. Just read up on it.
also depends how secure they want it and what you have access to on their server you can do it: js - not very secure php - secure but needs mysql asp - same as above htaccess - from what i've read this is the most secure method but you need access to apahce config files which you may or may not have access to