Created by Q as both a test and a form of amusement, the Gauntlet is a place to compete against other Captains by pitting your Crew's skills against theirs, and determine who among you are best equipped to handle the current crisis.
When entering the Gauntlet, you will select a team of five Crew members that are broadly qualified to handle a variety of challenges. These tests will focus on your Crew's ability to adapt--a value range known as "Proficiency"--rather than their core skills themselves. You'll see this range in parentheses next to the core Skill value when looking at a particular Crew member's abilities. Select a team that possesses a a wide variety of Skills, all of which are highly adaptable, pay special attention to the "Featured" Skill which will be challenged more frequently, and be aware that once you've made your Crew selection, you won't be able to swap anyone else in until this round of the Gauntlet ends.
A round in the Gauntlet lasts roughly 48 hours. During that time, you will need to return periodically to engage new opponents and accumulate trophies, which are the basis of your final Rank. Engagements will always test two Skills at a time, so try to pick a Crew member who is proficient in both. If you are able to win several engagements without a loss in between, you can also develop a Streak that will yield additional opportunities for reward. Your team will become fatigued over time if you engage many opponents over a short period of time, and if one of them fails an engagement, they will need to bow out for a time to reflect and recover. If your team has become too worn down to continue, let them rest and check back later.
The ultimate goal of the Gauntlet is to win the favor of a very unique individual who might join your Crew. There is no sure way to accomplish this, as all is subject to Q's whims, but achieving a higher Rank or accumulating a long Streak will give you better odds of drawing their attention. The individual in question will occasionally change, so if you have your heart set on the one currently available, compete well and often for the best opportunity to recruit them before they move on. Esteemed individuals such as Locutus of Borg and Bartender Guinan have been among those available, and there will be others to come.
To access the Gauntlet, tap the bottom-right button vaguely resembling Judge Q's head. The Gauntlet becomes available at level 10.
Note that the Gauntlet utilizes theoretical copies of the Crew that you select upon entry. This means that any changes made to one of those Crew members, such as leveling up or equipping Items, will not be reflected in that round of the Gauntlet. It also means that Dismissing or Freezing them won't remove them from that Gauntlet round, and that you will be able to use the real version of them in other types of Missions at the same time.