1To move toward or arrive at the speaker's location or time — the opposite of 行く. Used for both physical motion (someone approaching) and abstract arrival (seasons, occasions, news, etc.).
電車がもうすぐ来る。
The train will be here soon.
明日、お客さんが家に来る予定だ。
A guest is scheduled to come to my house tomorrow.
春が来ると、桜が一斉に咲き始める。
When spring comes, the cherry blossoms all start to bloom at once.
メールがまだ来ていない。
The email still hasn't arrived.
2Auxiliary use after a verb's -te form: '〜てくる' meaning to go (somewhere), do the action, and return. E.g. 買ってくる = go and buy and come back. Typically written in kana in this auxiliary use.
ちょっとコンビニに行ってくる。
I'm just going to run to the convenience store.
お土産を買ってきたよ。
I brought back some souvenirs for you.
駅まで友達を迎えに行ってきます。
I'll go pick up my friend at the station and be right back.
3Another auxiliary use of '〜てくる' after a verb's -te form: expresses a gradual change toward a state (暖かくなってきた = it's been getting warmer), or continuation of an action from the past up to the present (頑張ってきた = I've been working hard until now). Usually written in kana.
だんだん寒くなってきた。
It's gradually getting colder.
雨が降ってきたから、傘を持っていこう。
It's started to rain, so let's take an umbrella.
ずっとこの仕事を続けてきたが、そろそろ辞めようと思う。
I've kept doing this job all this time, but I'm thinking it's about time to quit.
4In the pattern '〜から来る' or '〜から来ている', to come from, originate in, or be caused by something.
この習慣は中国から来ている。
This custom comes from China.
頭痛はストレスから来ているのかもしれない。
The headache may be coming from stress.
「カラオケ」という言葉は「空のオーケストラ」から来ている。
The word 'karaoke' comes from 'kara (empty) + okesutora (orchestra)'.
5In the colloquial pattern '〜と来たら', used to introduce a topic and then comment on it — often with exasperation, complaint, or emphatic opinion. Roughly 'when it comes to ...' or 'as for ...'.
うちの息子と来たら、毎日ゲームばかりしている。
As for my son, all he does is play video games every day.
最近の天気と来たら、まったく予測できない。
The weather lately — it's just completely unpredictable.
数学と来たら、彼に勝てる人はいない。
When it comes to math, there's no one who can beat him.