会: ability from learning
会 means you acquired the skill through practice or study. 我会游泳 (I can swim — I learned how). 我不会打网球 (I cannot play tennis — I never learned). If the question is whether you know how to do something, the answer uses 会. 会 also carries a second meaning of future-likelihood: 明天会下雨 (it will probably rain tomorrow), but the modal-ability meaning is the primary HSK 2 sense.
能: current capability or possibility
能 means the conditions right now allow it. 我今天能来 (I can come today — I am available). 我能吃辣的 (I can eat spicy food — my body handles it). 明天会议能推迟吗 (can the meeting be postponed tomorrow — is it possible). 能 often involves context, circumstances, or physical capacity. Someone who knows how to drive (会开车) might still not be able to drive right now if they are too tired (不能开车).
我会开车，但是今天太累了，不能开。
Wǒ huì kāichē, dànshì jīntiān tài lèi le, bù néng kāi.
I can drive (I learned how), but today I am too tired, I cannot drive (right now).
可以: permission or polite request
可以 expresses permission. 你可以走了 (you may go). 我可以用你的电脑吗 (may I use your computer). In positive contexts 可以 and 能 overlap (both can express "it is OK to..."), but when you are asking for permission, 可以 is the polite default. In HSK-level speech, 可以 is the form used on signs and polite requests: 请坐 (please sit), 您可以在这里坐下 (you may sit here).
Overlap and decision rule
When in doubt: 会 = I have learned this skill; 能 = I am able to do this in current conditions; 可以 = someone has given or could give permission. One useful test: replace "can" with "know how to" → 会; replace with "am able to" → 能; replace with "am allowed to" → 可以. If multiple fit, pick the clearest intent.
Negation
All three take 不 for plain negation: 不会, 不能, 不可以. 不会 = do not know how. 不能 = cannot in current circumstances. 不可以 = not allowed. Signs and rules use 不可以 or the stronger 不许: 这里不可以抽烟 (no smoking here).
Common mistakes
"I can speak three languages" = 我会说三种语言 — use 会, not 能, because you learned them. "Can I open the window?" = 我可以开窗吗 — permission, not skill. "I cannot come tomorrow" = 明天我不能来 — circumstances, not skill. Mixing them up is intelligible but sounds off.