1To be (somewhere); to exist. Used for animate beings — people and animals — paired with ある, which is used for inanimate things. Almost always written in kana; the kanji 居る feels old-fashioned in modern writing.
公園に小さな犬がいる。
There's a small dog in the park.
弟は今アメリカにいます。
My younger brother is in America right now.
部屋には誰もいなかった。
There was nobody in the room.
池に金魚が三匹いる。
There are three goldfish in the pond.
「先生はいますか？」「今、職員室にいますよ。」
"Is the teacher in?" "Yes, she's in the staff room right now."
2To stay; to remain in a place. An extension of sense 1 with a stronger emphasis on duration or intent ("I'll stay home," "please stay here").
今日は一日中家にいる予定だ。
I'm planning to stay home all day today.
雨だから、しばらくここにいよう。
It's raining, so let's stay here for a while.
戻るまでここにいてください。
Please stay here until I get back.
母は来週まで実家にいるそうだ。
I hear my mother will be staying at her parents' place until next week.
3An auxiliary verb attached to the -te form to indicate that an action is in progress (the progressive aspect: "is doing ~"). One of the most important grammatical patterns in Japanese. With activity verbs (食べる, 走る, 書く, etc.), it expresses the action being underway. The form is identical to sense 4 (resultant state) — only the interpretation differs.
今、本を読んでいる。
I'm reading a book right now.
弟は毎日ピアノを練習している。
My younger brother practices the piano every day.
雨が降っているから、傘を持っていったほうがいい。
It's raining, so you should take an umbrella with you.
母は台所で料理を作っている。
My mother is cooking in the kitchen.
子供たちが公園で楽しそうに走っている。
The children are running around happily in the park.
4An auxiliary verb attached to the -te form to indicate a state resulting from a completed action or change (resultant state). The form 〜ている is identical to sense 3, but with punctual/change-of-state verbs (結婚する, 開く, 死ぬ, 着く, etc.) it expresses the lasting state after the change rather than ongoing action.
ドアが開いている。
The door is open.
田中さんはもう結婚している。
Tanaka is already married.
父は東京に住んでいる。
My father lives in Tokyo.
私はその人を知っている。
I know that person.
電気がついているから、誰か部屋にいるのだろう。
The lights are on, so someone must be in the room.