Basics of Current Electricity I have certain questions which confuse me a great deal. I want to have a very basic and clear idea about some concepts. Here it goes:
1) I am really confused on how a battery actually works. The very basic question that I have is whether positive ions flow in a battery? I ask this because my textbook uses phrases like 'positive ion flow'. While explaining the working of a battery, it highlights the journey of a positive ion across the circuit. I get confused because positive ions don't move in a wire. 
2) I know potential difference between two points is the amount of work that needs to be done to move a positive charge from one point to the other. But when we talk about a battery, my textbook uses the phrase 'Potential difference between the electrodes'. I don't quite understand what this means. Does this mean that one electrode of the battery has an excess of electrons and the other has deficiency? What is the path of a charge through a cell?
3) Also, what does 'Potential Drop' means? Why does the potential drop across a resistor? I tried reading some resources but I don't get it.
4) What is EMF? I know the basic definition, but can you please explain with simplicity? 
I don't know if I'm unnecessarily confusing myself? Perhaps I should accept some things as facts. The main problem is that I really get confused while solving a problem. Clearing the questions I asked above would make me understand what I actually am doing while solving a question. Thank you! 
 A: Regarding questions one and two, I recommend that you search up how a battery works. But in essence, ions flow in the battery, whilst electrons are routed through the wire to do useful work. This is possible through the chemical energy contained in the battery.
Instead of uttering "potential difference" as one phrase, I find it helpful to emphasize `potential', as in "potential difference". Think about two books of the same mass, one lying on a table and the other on a tall shelf. The latter will have higher potential because it resides higher - the difference in the potential of the two is called potential difference.
As for question three, potential is lost invariably due to resistance. If we consider the analogy of the water pipe it is easy to understand a pump can increase the potential; likewise a treadmill can reduce it. Note that throughout the wire the potential is dropping, but usually we consider the resistance to be negligible.
EMF can be thought of to be the same as the "voltage" of a battery, except that it accounts for the internal resistance as well. Recall that EMF and potential difference are linked by $\mathcal{E}=V+Ir$. At its simplest, some potential is lost in the battery due to internal resistance $r$.
A: I'll briefly address several of your questions but do consider breaking up these questions into separate posts if you want greater detail.

The very basic question that I have is whether positive ions flow in a
  battery?

Inside a battery (cell) is an electrolyte that reacts with the electrodes, removing electrons from one and adding electrons to the other.  An electrolyte is a solution containing electrically charged atoms.
When an external circuit is connected to the battery, electrons will flow from the terminal with excess electrons, through the external circuit, to the terminal with a deficit of electrons.
Within the electrolyte, there is an electric current due to the flow of ions.  Within the terminals and external circuit, the electric current is due to the flow of electrons.

Does this mean that one electrode of the battery has an excess of
  electrons and the other has deficiency

That's correct.

Also, what does 'Potential Drop' means?

Essentially, electric charge moves from higher potential to lower potential within the resistor thus, "potential drop".  Within the battery, electric charge moves from lower potential to higher potential thus, "potential rise".

What is EMF?

In this context, emf can be thought of as a source of electric energy.  The battery, in essence, converts chemical energy to electric energy.  The strength of emf is measured in volts and is numerically equal to the open circuit voltage across the battery terminals.
Note that if one places a short circuit across the battery terminals, the voltage across the battery becomes zero yet there is a large current circulating through the short circuit.  It is the emf of the battery that 'drives' this current through the short circuit. 
Quoting from the Wikipedia article "Electromotive force":

A source of emf can be thought of as a kind of charge pump that acts
  to move positive charge from a point of low potential through its
  interior to a point of high potential. … By chemical, mechanical or
  other means, the source of emf performs work dW on that charge to move
  it to the high potential terminal.

A: 1)  Different batteries may operate differently.  But yes, it may be much different from a wire.  Wires are solid materials.  In a solid, the atoms (and therefore the positive nuclei) don't move.  So the only charge carrier in the wire are the negatively charged electrons.
But a liquid electrolyte can have positive ions inside, and those ions can move in the liquid environment.  But that depends on the chemistry.  As an example, your standard alkaline battery communicates a hydroxide ion $[OH^-]$ which is negative.  In lithium-ion batteries, the mobile charge is the positive lithium ion $[Li^+]$.
2) Yes, the chemistry is such that electrons are produced on the negative electrode.  This chemical process continues until enough electrons are there that an electric field is produced that gives the electrode a lower potential than the other electrode.  When this happens, the chemical reaction is no longer "strong" enough to push more electrons onto the electrode.  The chemistry and the electrostatic forces are now in equilibrium.  
Attaching the cell into a circuit disturbs this equilibrium.  The electrons can leave the electrode, diminishing the electric field strength, so the chemical reactions can now deposit more electrons there.
Net charge moves inside the cell by ions moving through the electrolyte.
3) Potential drop is simply the potential difference (voltage difference) between one side of an element (a resistor for example) to another.  If you think of taking a path beginning at the high side of the battery along the circuit, then you will always be descending (in potential) until you finally reach the battery.  In this way of thinking, you can imagine your potential dropping as you pass an element of the circuit.
4) EMF is the "force" that imparts energy to charges.  Generally, you simply think of it as the thing that is producing the voltage in a supply, and it is given a voltage value.  Have you looked over the wikipedia article or other sources and had a problem with them?  Perhaps asking about a particular issue would be better than simply asking for a definition here.
