能 indicates an ability (a form of expertise), or possibility.

他一天能记住五十个汉字 He can remember fifty Chinese characters a day.

一百镑能买到很多东西。 One can buy a lot of things with a hundred pounds.

Please note when used in a negative form or a question, 能 means 'possible to', 'can manage to'.

他病了，所以今天不能来上班了。 He's ill, so he cannot come to work today.

我不能帮您办信用卡。 I cannot give you a credit card.

您能不能帮我办一张信用卡？ Could you provide me with a credit card please?

我怎样才能开个人账户呢？ How can I open an individual account?

会 means 'to be able to'. It is specifically used to indicate an acquired ability, knowing how to do something after some learning or practice.

莉莉会说一点儿中文。 Lilly can speak a little bit of Chinese.

我不太会打乒乓球。 I don't really know how to play table tennis.

你会开车吗？ Can you drive?

Please note that 会 is often used to express a definite prediction, in which case it can be translated as 'will'.

我会照你的意思办。 I will follow your recommendation.

他今天不会来了，他去外地出差了。 He won't come today, he is on a business trip.

明天会下雨吗？ Will it rain tomorrow?

可以 mainly indicates permission.

你们可以进来了。 You can come in now.

您不可以透支。 You cannot be overdrawn.

我可以走了吗？ May I leave now?

Please note that:

• When used in a positive sentence, 可以 also indicates suggestion.

您可以去分行办理新卡。 You can go to a branch to apply for a new card.

你有什么问题都可以问我。 You can ask me if you have any questions.

• When used in a question, 可以 also indicates the expectation of a positive answer.

你可以帮我一个忙吗？ Could you do me a favour?

你可以把这本书借我看看吗？ Could you lend me this book?