能 as capability, 可以 as permission
能 covers physical or circumstantial ability: 我能游泳 (I can swim, I know how), 今天我能加班 (I can work overtime today, circumstances allow). 可以 covers permission or social acceptability: 你可以进来 (you may come in), 这里可以抽烟吗 (can I smoke here, meaning is it allowed?). The split is learned quickly: CAN-DO = 能, MAY-I = 可以. For overlap cases (能 often works where 可以 does), 可以 sounds slightly more permissive and polite.
我能游泳,但是在这个池子里不可以游泳。
Wǒ néng yóuyǒng, dànshì zài zhè ge chízi lǐ bù kěyǐ yóuyǒng.
I can swim, but swimming is not permitted in this pool.
可 as a standalone
可 alone is rare in modern Mandarin; it mostly functions as a morpheme in compounds. When you do see standalone 可, it is often in literary or classical contexts: 可圈可点 (worthy of praise), 可歌可泣 (inspiring and tragic). In modern speech, 可 standalone as a modal is uncommon; use the compound 可以 or related forms.
Compound: 可以
可以 (kěyǐ) is the modern standard for "may, can, permitted". It is how you ask permission: 我可以用你的笔吗 (may I use your pen?), 可以坐这里吗 (may I sit here?). It also expresses mild capability: 这件事情我可以做 (I can do this task). Use 可以 as the go-to for permission questions and soft "can" statements.
你可以帮我一下吗?
Nǐ kěyǐ bāng wǒ yíxià ma?
Can you help me a moment?
Compound: 可能
可能 (kěnéng) means "possible, maybe" and covers likelihood (see keneng-vs-yexu). It is a probability marker: 他可能会来 (he may come). 可能 as a noun means "possibility". This compound comes from 可 (may) + 能 (able), literally "may-be-able". Learners recognize the parts but should treat 可能 as a lexical unit, not as modal stacking.
Overlap between 能 and 可以
In many sentences, 能 and 可以 are interchangeable: 你能帮我吗 (can you help?) and 你可以帮我吗 (can you help?) are both fine. The nuance: 能 slightly emphasizes capability; 可以 slightly emphasizes permission or willingness. In polite requests, 可以 is slightly softer. In claims about one's own ability, 能 is slightly more direct. Both are common; rarely wrong.