1To persevere and do one's best — to keep making a strong effort toward a goal even when it's hard. 頑張って / 頑張ろう (do your best / let's do our best) is one of the most common expressions in Japanese. Often written in kana (がんばって) for soft encouragement.
試験まであと一週間、最後まで頑張ろう。
One week left until the exam — let's give it our all to the end.
「無理しないで、でも頑張ってね」と母が言った。
「Don't overdo it, but do your best」, my mother said.
新しい仕事はつらいけど、もう少しがんばってみる。
The new job is tough, but I'll try to stick with it a bit longer.
運動会で一生懸命がんばる子どもの姿に、思わず涙が出た。
I couldn't help tearing up at the sight of my child trying so hard at the sports day.
ここまで頑張ってきたのだから、今あきらめるのはもったいない。
You've worked so hard to get this far — it'd be a shame to give up now.
2To insist on something or stick firmly to one's opinion or position — used as 〜とがんばる to describe someone who won't give ground.
彼は自分は悪くないと、最後までがんばった。
He insisted to the very end that he had done nothing wrong.
どんなに説得しても、彼女は行かないと頑張っている。
No matter how much we try to persuade her, she keeps insisting she won't go.
3To stay put and hold one's ground — to remain at one's post or position and endure, refusing to budge in a difficult situation.
主力選手が抜けても、チームは最下位にならずに頑張っている。
Even after losing its star player, the team is holding on and staying out of last place.
再開発の波の中でも、老舗の店主は店を閉めずに頑張っている。
Amid the wave of redevelopment, the owner of the long-established shop is holding out and keeping it open.