How do you do a jQuery Ajax call and authenticate the call prior to sending the request?
I have not logged in so have have to authenticate. Security is not an issue anyone can get access just need to authenticate. It just basic http authentication you can take a look at the API http://lighthouseapp.com/api/the-basics
If a user on your website is already authenticated, in most circumstances you don't need to do anything - auth cookie will get sent with your AJAX call. Otherwise, you can try HTTP Basic auth.
If you are trying to get javascript to do the authentication without any user interaction, don't.
Hardcoding your authentication logic into code available to the client could severely compromise the security of the API. If you are going to put the username/password into your javascript, why even use one at all?
If you have access to the API and can rework the authentication, you could try a tokening system for authentication. Just my $.02
Since you don't specify what kind of authentication you're using, I'm going to make some big assumptions that you have some sort of login page/action that you post the username and password to, using those as the parameter names. If you have other fields -- like hidden fields to prevent cross-site request forgeries, you'd need to include those as well. I'm also going to assume that you know you're not already authenticated. There are ways to detect this, but I'm not going to cover them. I'll further assume that you're posting to the web site's actions, not to some API that requires a separate type of authentication.
The first thing you'd do is generate a POST (I assume) to the login action with a correct username/password combination. How you get these is up to you. This will authenticate you with the web site and supply your browser with the appropriate authentication cookie to send with future requests.
You'll need to detect and handle an authentication failure. If your login action understands that it might be invoked via AJAX (using the HTTP_X_REQUESTED_WITH header is a good bet), then it can return JSON with a status setting otherwise you'll need to scrape the returned HTML to figure it out.
Once you have the authentication cookie, you should be able to make your actual AJAX request without any special handling.