Accessing common database for mobile app and website

first there are two parts:

Frontend and Backend

In your case you need and API written in the script/programming language you want (nodejs, php, python, perl, java, ...) as your backend.
But this should depend where your website gets the data from.

If there is a backend for your website or your website is written in php --> create a little php-api for your app.
If you have thousands of parallel users in your app you will reach the limits of simple php, perl, python apps really fast (without looking for loadbalancing and....).
And if you have an own server instead of a simple small webspace you can give nodejs a try

So the communication would be the following in general:

APP --> sends requests with angularjs $http-service or $resource --> API written in the language of your choice is connected to your database and receives requests --> gets data and processes data --> sends data to you app.

The interesting part is the getting and processing part of your requests.
There you should look what request-method you have like GET, PUT, POST, DELETE. and then you should send information within the request what you want to do with what.

like a example request:
GET http://xxx.xx/api/users
--> get all users of the db
GET http://xxx.xx/api/users/id/1234
--> get the user with the id 1234
DELETE http://xxx.xx/api/users/1234
--> delete user with id 1234

if you have a simple index.php file your urls can look like this:
GET http://xxx.xx/index.php?class=users&.....

so you can first decide what method the request has --> okay get... next step --> what class, data it belongs to --> oh there is the query paramerter "class" --> oh and the value "users" --> okay get all users
or this one:
GET http://xxx.xx/index.php?class=users&id=1234

get the user with id 1234.

Sometimes you need a special action because you have multiple get actions on the same data like users can have ratings and you want the ratings of a special user:
GET http://xxx.xx/index.php?class=users&id=1234&action=ratings

So the important point of an api is an unified url-scheme.