Going stealth means for a trans person to live completely as their gender identity and to pass in the public sphere. Most people are unaware of their transgender status. This doesn't mean their status is a secret to every single person; family and close friends may know.
For transexuals, going stealth is generally the goal of transition. Some transexuals and most genderqueer and bigender people purposely don't go stealth because they want the people around them to know they're trans. Some desire to go stealth but are unable to pass convincingly enough. Historically, going stealth is a very recent phenomenon since, for many people, hormones are necessary to pass.