In chemistry's voltaic cell, the anode (negative electrode) of the battery supplies electrons to the cathode (positive electrode) of the battery. In physics, the electrons flow from negative terminal (cathode) of the battery to positive terminal (anode) of the battery.
I don't understand this because in thermionic emission, electrons flow from cathode to anode too through an electric source (which I think is a voltaic cell).
I think you've confused the flow of electrons with the flow of current. Before it was discovered that electricity is the result of the flow of negatively charged electrons it was proposed that electricity result from the flow of a positive charge. We called this current and this convention has remained to this day.
So current is effectively the flow of positive charge not the flow of electrons, which are negatively charged. So current flows from the cathode to the anode, but electrons flow from the anode to the cathode. This is true in both physics and chemistry.
The anode is where oxidation takes place, the cathode is where reduction takes place.
Thus, in a battery the anode supplies the electrons that enter the circuit and the cathode accepts them. But in an electrochemical cell the anode is where anions are attracted to to be oxidised by electrons supplied by the power source.
This can be confusing.
I can sympathise with your situation; for reasons unknown to me, people are taught in ways that often leaves gaps where such questions/misconceptions originate.
In general, to avoid confusion it is best to define anode and cathode in terms of reduction/oxidation. The anode is the electrode where oxidation takes place, and the cathode is the one where reduction takes place.
Also, in an electrochemical cell, electrons don't flow, but ions do.
Now, let's address the differences between galvanic and electrolytic cells.
In a galvanic cell the reaction proceeds without an external potential helping it along. The oxidation reaction which produces electrons takes place at the anode, results in a build-up of negative charge at the anode in the course of a reaction till equilibrium is reached. Thus the anode is negative.
On the other hand, reduction takes place at the cathode, and consumes electrons, and thus leads to a build-up of positive charge (or depletion of negative charge). Thus the cathode is positive.
The situation is reversed in an electrolytic cell where you do provide an external potential, and you are essential pushing the reaction in the reverse direction (say, when you charge a cell). So while oxidation still takes place at the anode, the anode is now positive and while the cathode is still the site of reduction, it is now negative.
You can read more here
In physics, circuits usually depict a current which moves from the positive electrode of a batter to the negative one. This is not representative of the flow of electrons; it is a historical artifact of sorts, because this convention was agreed upon before the discovery of electrons. Read more here
Sorry but the confusion remains even on the web pages on the net. Chemistry sites tell us that the anode is the source of electrons and are attracted by the cathode. I just finished looking at the anode on a natural gas hot water heater which is magnesium rode. No current is applied. Mg is to the left on the periodic table and will give up electrons to form compounds. It is called an anode. This is true for batteries and chemistry.
Physics, however, specifically in electronics the anode attracts electrons and the cathode gives off electrons. The cathode ray tube is a case in point. The electronic gun is the source of electrons and the front of the old television screen are the anode. The negative of the battery is the cathode and the positive is the anode. Chemistry and physics are at odds with each other.
