We use CiviCRM for our nonprofit, and we're trying to figure out how we can export our raw data in order to upload it into some custom apps. Any help would be very much appreciated!
The answer rather depends on what data you have in CiviCRM and what you want to export.
One option is to export the data as a CSV file as Demerit says. This will allow you to export certain types of data - for example: contacts, event participants, activities. You can do this via advanced search. See the manual for more details: Exporting your contacts.
Another option is to access the data directly from the database. This gives you more control to access the data that you need but would require some understanding of the database structure and SQL. If you have access to phpMyAdmin you will be able to export the SQL data using this (log in to phpMyAdmin and click on the Export tab). Alternatively your website hosting company may have a tool that allows you to export the SQL - you could contact them for help.
A third option to export the data via the API which would require some programming.
Although it is possible to access the DB directly (it's just a MySQL database) that is not the recommended way to integrate with custom apps.
There are ongoing changes to the database tables which mean that even if your app integrates perfectly today, it may break tomorrow with the next release. The API tries hard to insulate you from underlying database changes.
Your local CS Dept should have little difficulty getting started with the API using the information here: https://wiki.civicrm.org/confluence/display/CRMDOC/Using+the+API and the built-in API explorer (see same doc) provides an easy way to experiment and learn.
(Adding some.text because it seems stackexchange wants at least 30 chars of text. Ignore this text and just follow the link.)