When you put plain copper into plain hydrochloric acid, nothing happens. Aluminum reacts with hydrochloric acid, it produces aluminum chloride and hydrogen gas. For the electrochemical cell, the two half reactions are (I think):
2H+ + 2e- = H2
2Cl- + 2H2O + Cu+2 = 2H2 + CuCl2O2
Then, the copper gets more negative and the aluminum gets more positive so the electrons flow from the copper to the aluminum.
Is this all that happens?