Do you know how to process regular HTML form submissions? Then you can also process Acrobat form submissions: There are several options for you to choose from, the most straight forward one is to select the HTML form submission on the submit button's properties dialog.
How you would store the data is up to you, there is nothing in Acrobat that demands a special way to store data, you can decide what's right for your application. This can be in a database, but it could also be just a plain old CSV file - however I would strongly suggest a database.
