Can has two forms - the form can, which is used when the action refers to the present, sometimes to the future and the form could, which is used when the action refers to the past; the latter is also used as the form of the subjunctive mood.

If you gave me some money, I could buy this book.
Если бы ты дал мне немного денег, я мог бы купить эту книгу.

If you have given me your notes, I could have passed my exam yesterday.
Если бы ты дал мне свой конспект, я мог бы сдать экзамен вчера.

The modal expression to be able can be used to replace the verb can when the action refers to the future (смогу).

I'll be able to give you my notes on Saturday.
Я смогу дать тебе мой конспект в субботу.

The verb can expresses:

  • Physical ability to perform an action.

    I can't walk anymore.
    Я не в состоянии больше идти.

  • Possibility due to circumstances.

    I quite free today and can join you.
    Я сегодня совершенно свободен и могу к вам присоединиться.

  • Possibility due to some skills, knowledge.

    We can't do such sums yet.
    Мы еще не умеем решать такие задачки.

    Can any of you play the piano?
    Кто-нибудь из вас умеет играть на рояле?

    Note: Failure to perform an action can be expressed by the verb to fail + to-infinitive or by the modal expression not to be able.

    I failed to meet him on Saturday.
    Я не смог с ним встретиться в субботу.

  • The verb can, like the verbs may and must, processes secondary modality, - it can express incredulity, doubt, astonishment with all forms of the infinitive in negative and interrogative sentences. (Не может быть! / Неужели!)

    Can he be right?
    Неужели он прав?

    He cannot know about it.
    Не может быть, чтобы он знал об этом.

    Can she be still waiting for him? She cannot be still waiting for him.
    Неужели она все еще ждет его.

    Can he have emigrated?
    Неужели он эмигрировал.

    He cannot have emigrated!
    Не может быть, чтобы он эмигрировал.

    Can they have been waiting for us since the very morning?
    Неужели они нас ждут с самого утра.

    The forms can and could are interchangeable here because the time of the action is expressed not by the modal verb but by the form of the Infinitive.