Indeed if we are going to talk about COULD-CAN and ABLE TO we would write an entire book on this subject.

   Thus, to make a long history short:

    1-COULD is used as past tense of CAN (in indirect speech)

     ex: He says he can speak English well. --> He said he could speak English well.

     2-COULD can refer to past physical or intelectual abilities (though, these habilities may or may not exist in the present)

    ex:  When I was in senior high school, I could understand math well.

             (not necessarilly means that I am not a skilled at math now, but people usually

  assume that this ability no longer exist)

           When she was younger, she could dance better than anyone.

              (same above remark)

         3-COULD is used to ask something to somebody (to make a request).

         4-COULD is used in present, past , and future.

         5-CAN is used only in present and future.

         6-(be) able to...  is possible instead of can, but CAN is more usual.

         7-As CAN has only two forms, be able to provides the way to refer to abilities in other tenses:

          ex: I can´t sleep.   (present)

               I haven´t been able to sleep recently (present perfect)

               Tom can come tomorrow.

               Tom might be able to come tomorrow.  (can has no infinitive)

           8-Still, as for CAN vs WAS ABLE TO; sometimes CAN is the past of can.

            You use could especially with:     see, hear, smell, taste, feel, remember, understand.

           9-COULD, to refer to permission to do something.

            ex: We could do what we wanted.

           These are some aspects, but you should see more or in books or sites or Forums.

            Good grammar books on the subject:

            1-Michaelis inglês-gramática aplicada ed. Melhoramentos

            2-English Grammar in Use -a self study reference and practice book for intermediate students, with answers.  Author: Raymond Murphy -Cambridge Univ. Press.

           Lots of luck, see you later.