Can is used to say that something is possible or allowed or that somebody has the ability to do something.

After can we use the base verb.

eg. She can play baseball. / Can you read?

The negative form is cannot / can’t.

 

Be able to means the same thing as can, but can is more usual.

eg. She is able to see the mountains from her hotel room.

 

The difference between the two is that can has only two forms: can and could  (can is used to talk about the present while could can be use to talk about the past) while be able to can be used in all verb tenses.

 

Could is used for general ability an in polite requests.

eg. She could speak Chinese.

Could you please, pass the salt?

Was able to / managed to on the other hand is used to say that somebody did something in a specific situation.

eg. He managed to finish the project on his own. (not could)