A counter for occurrences — used to count the number of times something happens (e.g., once, twice).
この映画はもう三回見た。
I've already seen this movie three times.
一回だけチャンスをください。
Please give me just one chance.
彼は大会で何回も優勝している。
He has won the tournament many times.
A time or instance — referring to a particular occurrence or occasion (used as a noun: 'that time/instance').
彼に会ったのはその一回だけだ。
I met him only that one time.
その回は特別ゲストが出演した。
That episode featured a special guest.
An inning/round — used in sports (e.g., baseball innings) or rounds of play.
試合は五回裏に逆転された。
The game was turned around in the bottom of the fifth inning.
三回の表にホームランが出た。
A home run was hit in the top of the third inning.
Used colloquially as a suffix meaning an episode or installment (often for TV shows or serialized content).
この回のテーマは友情だ。
The theme of this episode is friendship.
先週の回を見逃したので、今夜もう一度見るつもりだ。
I missed last week's episode, so I plan to watch it again tonight.
Used in compounds referring to the Hui people (回族) — the kanji appears in the name of this Chinese ethnic group; not commonly used alone in daily Japanese.
中国の回族（かいぞく）はイスラム教を信仰する民族の一つだ。
The Hui (回族) in China are one of the ethnic groups that follow Islam.
An older term for Islam (回教). Nowadays, 'Islam' (イスラム教) is the standard and preferred term.
『回教』という表現は古い言い方で、現在は『イスラム教』と呼ぶのが一般的だ。
The term '回教' is dated; nowadays it's more common to say 'Islam' (イスラム教).