If cross-browser requests are forbidden, how do I enable API access?

[HelpfulChad]

I am writing a C# web service that serves data from a back-end web server. If I want to let other users query this service using client-side code, how would I provide them with API access? For instance, currently the web service is on Domain1.com. Now, if I want to let someone from Domain2.com access my API using jQuery, how would I do it?