To raise one's hand (e.g. to ask a question or indicate a desire to speak in class/meetings).
授業中に質問がある人は手を挙げてください。
If you have a question during class, please raise your hand.
朝のミーティングで、発表したい人は手を挙げた。
At the morning meeting, those who wanted to present raised their hands.
ボランティアを募るときは、興味がある人は手を挙げてください。
When recruiting volunteers, please raise your hand if you're interested.
To surrender; to give up resisting (often used idiomatically).
包囲されて犯人はついに手を挙げた。
Surrounded, the criminal finally surrendered.
長時間の議論の末、彼は反論を諦めて手を挙げた。
After a long argument, he gave up his objections and conceded.
To raise one's hand toward someone as a threat to strike (a sign of intent to use physical force).
子どもが弟に向かって手を挙げたので、母親がとっさに止めた。
A child raised a hand toward his younger brother, so the mother quickly stopped him.
彼は怒って相手に手を挙げたが、誰かに腕を掴まれて止められた。
He angrily raised his hand at the other person, but someone grabbed his arm and stopped him.
To improve; to get better at something (idiomatic, akin to 腕を上げる).
毎日練習した結果、彼女は手を挙げたと周囲が驚いた。
After daily practice, those around her were surprised at how much she had improved.
料理教室に通ってから、彼の腕は上がり、手を挙げたと評判だ。
Since attending cooking classes, his skills have improved and people say he's gotten much better.
経験を積むにつれて、自然と手を挙げる場面が増えた。
As he gained experience, occasions where he stood out (improved) increased.
To volunteer or indicate intention to participate (to raise one's hand to join or sign up).
新しいプロジェクトに手を挙げたのは私だ。
I'm the one who volunteered for the new project.
クラスで代表を決めるとき、数人が手を挙げた。
When choosing a class representative, several people raised their hands.