"Can" is a modal which is used either to indicate Ability or to indicate Capability/Skill. It is ambiguous in the sense that one's Ability to do something might be compromised by
(i) a lack of knowledge or experience
(ii) the prevailing circumstances.
Thus, in isolation the question "Can Tom use the computer" could be taken to mean
(i) Does Tom have the knowledge and background that would confer upon him the capability or skill to use the computer
(ii) do present circumstances permit Tom the use of the computer. (eg. time; spare computers etc)
The question "Can I use a computer" would usually be taken to have meaning along the lines of (ii), because one would expect a person to know whether they themselves have or don't have a certain capability or skill.
The question "can you use a computer" could be easily taken to have either meaning, especially if the listener knows that the speaker has no prior knowledge as to the listener's abilities or skills. In fact in English one must occasionally seek clarification from a speaker because of this inherent linguistic ambiguity in English.
Your teacher should have made his jokes only after he had made you aware of this important linguistic point. My guess is that he was not aware of the idiosyncratic subtleties of Modality in English or of the importance that there is for teachers in finding ways to make learners aware of them. So yes, of course English speakers understand "Can you...?" questions, and they often can be confusing even for native speakers. Your teacher simply wasn't doing his job.
If you start your question with "May I..." "Could I...." Please can I...." and so on, not only is it more polite, it also immediately removes the ambiguity.