I have built a simple test API for a CakePHP application that will let a user login from a mobile device (or any device for that matter) and get a JSON response. This API co
Make your app login everytime, but not with login-pass pair as Swayok lastly suggested. When you login, server generates a token and returns it back to the client. Client then uses this token whenever it makes a request. On each request, server checks whether the token is valid and if so, executes the request.
This is very similar to how sessions work in that, server side frameworks manage it internally and these tokens expire from time to time. However, as Swayok rightuflly pointed out, you don't want session mainly because you're RESTful API should have no state. You get the same utility without storing any user specific data regarding user and logging user in with every request.
Here's a good article on this, or you can try the Facebook Graph API explorer to see it in action