The easiest way would be to store your dates as timestamps, then you could subtract the current timestamp with the one you've saved in a database.
The PHP function time() returns the current timestamp – that is the number of seconds since the 1st of January 1970. You can then format a timestamp $stamp to your liking with date('m/d/Y', $stamp), for example.
Aside from facilitating arithmetic operations with dates, you can display more or less information with timestamps by formatting them with date(), as well as show different formats (July 13, 2014 vs. 07/13/2014). If you save a date as a string, e.g. "8/10/2014", it will be very complicated for you to change the format, and it won't be possible to get the correct time, for example.
Finding how long ago in days a timestamp $stamp was to the current time is very easy:
$now = time();
$days = ($stamp - $now) / (24*3600);
Use round() to get a full number if desired (e.g. 7.2309 would simply become 7).