As a slang term, it is popularly used in the Black community--though apparently Tila Tequlia (herself not an activist) uses it as well. Though the phrase "stay woke" was particularly popular in 2014, Erykah Badu used it in her 2009 song "Master Teacher."
The most popular and frequent usage is "stay woke." This NYMag article tried "wokest."
Usually the awareness would be toward a socioeconomic or political situation, like the Black Lives Matters movement. (And not a batshit conspiracy that the world is flat.)
Don't fight it. And unless you're Black, don't say it. You'll look fake.
Edited to add this video, where DeRay McKesson explains woke to Stephen Colbert:
What does "I stay woke" mean?, Yahoo Answers
Stay Woke: All the New Slang of 2014 in One Video, Complex
Stay woke, Urban Dictionary
woke, Urban Dictionary