do your best; go for it; hang in there
A common encouraging interjection used to cheer someone on, similar to 'good luck' or 'you can do it'.
試験、頑張ってね。
Good luck on your exam.
明日の試合、頑張って！
Go for it in tomorrow's match!
頑張って has a softer, more polite request-like feel, while 頑張れ is the blunt imperative form used bluntly among close friends or as a rallying cheer.
The te-form of the verb 頑張る (がんばる), which originally meant 'to insist on one's own opinion' or 'to be stubborn', later shifting to 'to persevere'.