How Long is a Submarine Day?
After 2012: In 2012 the Navy began to allow for 24-hour watch schedules for submarines by relaxing rules related to the 6-hour watch which had led to 18 hour days in practice. By 2014 24 hour schedules were mandated. This means although the fact about the Navy using 18 hour days was true, the traditional 18 hour day is no longer practiced. Instead, a 24 hour day with 8 on / 16 off is common and partial watches are allowed.
TIP: This article was written before the switch had fully been implemented. The changes have been reflected above, the rest of the article is about the 18 hour day.
Why Does the Navy Use 18 Hour Days on a Submarine?
The 18-hour day is a matter of practicality; it allows submarines to be on watch 24/7 with the space limitations of the submarine taken into account.
What Does Hours “Off” Mean?
A crew member’s “off” time is the time they are off the watch. They do a number of things during this time including drills, maintenance, and sleep.
Has it Always Been an 18-Hour Day?
The 18-hour day was created in the 1960’s to address the fact that in modern times submariners were suffering from sleep deprivation on long patrols due to the need to do things like train and conduct drills on their 8-hour off period. Scientists are showing that 18-hour workdays have a positive effect on submariners. 
This video can help you understand the history of submarines. Understanding the history of submarines helps understand why keeping an 18-hour schedule can make a difference over the duration of patrols.
How Do We Define “a Day”?
When we say submarine days are 18-hour days we aren’t referring to the definition of a day as 24 hours, we are referring to the definition of a day in a more broad sense and as a workday or schedule. We are referring to the way a day is experienced by a given group of people, rather than a day as a relation to the sun. 
The 18-hour day can also be referred to as a watch system or watch schedule.