Basically when you look at an anion gap, you're comparing between positive charges and negative charges. There are a whole lot of them on each side, sodium, potassium, calcium, magnesium etc. are positive and for negatives you have bicarb, chloride, sulfates, phosphates, proteins. But to calculate the anion gap, you use Na - (Cl + HCO3) because these are the major ones so the other ones don't contribute as much. Your body is in homeostasis so you will be at about the same level of the various ions in your blood unless something major happens. That equation is an artificial construct but people use it because it works well, most of the charges in your blood are due to those three ions. The rest of them are folded into an "anion gap" which is usually around 8-12 mEq/L (meq is like milli-moles but with charges so a mole of na is an equivalent of na because it has a +1 charge. a mole of calcium is 2 equivalents of calcium because it has a +2 charge)
So the way I think of it is when you have metabolic acidosis, either you have too many H+ or too few HCO3-. If HCO3 goes down and there is no increase in Cl, then you have an increased anion gap without an increase in chloride. This tends to happen when your serum hydrogen ions go up. H+ and HCO3- combined leads to H2CO3 which gets blown off as CO2 and H2O. Why would your hydrogen ion level increase? Because you're adding hydrogens to the system. What are the most common causes? You eat something with a lot of hydrogen (salicylates as acetylsalicylic acid, propylene glycol to pyruvic acid, methanol to formic acid); you make too much hydrogen endogenously (diabetic ketoacidosis making too many ketoacids, lactic acidosis because you're exercising too much or you took too much metformin); or you are not getting rid of hydrogen as in renal failure leading to uremia. Hydrogen goes up, HCO3 goes down but Cl stays the same so you have a "normochloremic" metabolic acidosis.
Hyperchloremic is basically when the charge difference (anion gap) from your equation: Na - (Cl + HCO3) stays the same when you're losing HCO3 because you're replacing it with Cl-. The way I think of it is when your RBCs are taking in CO2, they exchange Cl- (put it out into serum) for HCO3 (putting it into the cell) when it's in the pulmonary capillaries using carbonic anhydrase to make CO2 in the lungs to get rid of it. This bicarb/chloride exchange is what happens in the alpha intercalated cell for example in the collecting tubule. So normally you would get H2CO3 that turns into H+ and HCO3- in the intercalated cell in the kidney. You throw out the H+ into the collecting tubule to get peed out and you will reabsorb the bicarb by exchanging with chloride. In type 1 renal tubular acidosis, that doesn't happen so your bicarb levels go down and you hold onto the chloride. High chloride, low bicarb, same amount of negative charge -> hyperchloremic metabolic acidosis. You can also have this kind of exchange in the gut (diarrhea = loss of bicarb, retention of chloride).
Another way to think of it is you add saline to someone's blood through an IV. Na and Cl go up by definition and the concentration of HCO3 has to go down from the dilution. Automatically you have hyperchloremic metabolic acidosis (since you're adding Cl but not HCO3). If you think the H+ is going down as well, realize that you have volume overload which will decrease renin, decrease angiotensin 2, decrease aldo which decreases Na+ reabsorption and increase K and H reabsorption in the kidney as one mechanism (opposite of contraction alkalosis).
Type 1 RTA: Your alpha intercalated cells don't work so your h+/k+ exchanger doesn't work. You hold onto H+ and lose k+ in the urine leading to hypokalemia and acidosis. Also your urine is less acidic because you're not putting H+ into it. (high ph urine can lead to calcium phosphate stones)
Type 2 RTA: Your proximal tubules don't work so you lose HCO3 as well as phosphate, glucose, amino acids, urea, protein. Getting rid of HCO3 in the body means that you are acidifying the blood. The alpha intercalated cells are working fine so they push out hydrogen ions and you acidify blood. I don't understand this very well but I'm guessing that you're getting rid of more bicarb that you are getting rid of hydrogen but the buffering is sucking because you're also losing other acids like uric acid, amino acid.
RTA 4: not really a renal tubular acidosis because your kidneys are doing fine but basically your aldosterone level is low because of maybe your channels are crappy in the collecting duct or your aldosterone isn't working (spironolactone, aldo resistance, addison) and then you get high potassium (hyperkalemia). The acid in your blood goes up a little since you're not getting rid of H+ and K+ and the urine is acidic because high k+ leads to lower ammonia (not sure why) and ammonia buffers hydrogens in the urine so you get acidic urine.
