What do we mean exactly by the positive/negative charges in cathode and anode? In batteries, what exactly do we mean by negative and positive charges? My understanding is that the negative charge of the anode is basically an atom with an extra electron in the last orbital and a minus electron in the last orbit of the positively charged atoms in the cathode? And so the electrons keep flowing and interchanging because the atom is trying to reach a stable state. Is this correct or am I wrong?
If my understanding of positively/negatively charged atoms is correct, what do we mean by saying that the electron in itself has a negative charge?
 A: You observe that protons and electrons have opposite charges.  It is a matter of definition which is positive, and the convention was made before anybody knew about protons and electrons-I have heard it was on the basis of a static charge.  
It is the cathode that is negative relative to the anode.  For conductors, the conduction electrons are free to move and not tied to a particular atom.  But it is an excess of electrons on the cathode that makes it negative and a deficit of electrons on the anode that makes it positive.
A: Reduction (gain of electrons) at the cathode, positively charged in a battery providing energy, negatively charged at an electrolyzing surface (e.g., something being plated with metal) consuming energy. Oxidation (loss of electrons) at the anode, negatively charged in a battery, positively charged at an electrolyzing surface (e.g., the metal source electrode for the thing being plated).  REDCAT and LEO to remember it.
An electrochemical cell has separated internal reactive electrodes bathed in electrolyte that  through chemical reaction create the electric potential across their external poles.  The "charges" are extra electrons or holes in the band structure of the conductors.  There is a lot of slosh in a conduction band compared to a discrete atom ionizing.  There is little charge flow to equilibrium voltage.  Connect the poles and current flows, limited by external and internal resistance, bracketed by reaction kinetics at the electrodes.
IMPORTANT!  Current flow is opposite to electron flow (the negative sign in their charge).
An electron is a stable elementary particle that has measurable properties - mass, charge, a half unit of quantum angular momentum ("spin"); other observables derived therefrom. What an electron is, structurally, is undetermined as it appears to be a zero-dimensional point.  Ask your instructor if an electron neutrino is an electron without its charge.
