I'm assuming what you mean is that visitors to another site will make client side requests to your API, and you want to attribute these requests to that site.
There are several ways to go about this depending on how strong you need the authentication to be.
If you're just using the data for statistical purposes and maybe to stop unauthorized use from very unsophisticated attackers then you can use the HTTP referrer. This can of course be forged, so this won't be suitable if you require very strong authentication.
Another option is to use an API key / site ID that is passed along with all requests. An attacker could easily take the API key from one customer and abuse it, or use it on another site - but at least you can reissue the key/ID if it's abused unlike the referrer.
If you need very strong authentication then you'll have to have your customers generate a time limited signed token on the server side which the user then passes in their request to you. Basically, each customer would have a secret key on the server side and they'd use that to encrypt a message including a time limit - you can check that by decrypting it when the user passes it to you. You might be able to leverage something like http://jwt.io/ to do this. The disadvantage of this is that you can't do this completely on the client side, so your customers might have to make some changes to their application.