Display my DB queries on other websites

What can I make to allow people to display data from my own website's database on their website using a query?

For example if I have a table named "movie_quotes" which is comprised of various movie quotes.. How can other people query them on their own sites by either genre or dates. For example they can modify the "category" parameter to get quotes from comedy movies. http://www.mysit开发者_开发问答e.com/movies/?category=comedy

..and would it be possible for them to submit their own movie quotes from their own site into my DB?

I think this is related to API but I really don't know where to start.

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Here is a link to an open source PHP web services framework that might be useful. http://wso2.com/products/web-services-framework/php/

An API is the best way to serve the data so you have the ability to limit access, monitor it, and turn it off when needed. You can issue credentials as desired and still maintain security.