Household vinegar contains, among other things, an ingredient known as acetic acid, which gives it its characteristic sharp taste and strong odour. Interestingly, though, acetic acid is highly corrosive, and may cause severe burns to skin tissue, eye damage and damage to many types of metals. The reason why, in spite of this, vinegar is safe to consume is simply due to the concentration of the acid itself. In fact, in most cases, the concentration of an acid is the main indicator of its danger (to some extent). The most convenient way to express this concentration, and concentrations in general, is as a ratio of the moles of the acid, or 'solute', to the volume of the solution.
What is Molarity
The molarity of a solution is defined as the number of moles of a solute, like salt, divided by the volume of the solution (i.e. the saltwater itself). While the SI unit-derived form of molarity would technically be $mol/m^3$, the far more common form used throughout laboratories is $mol/dm^3$, which is often expressed in the equivalent form $mol/L$.
The Formula for Molarity
As should be relatively obvious from the description overhead, the mathematical formula for molarity is simply the ratio of solute moles and solution volume, which we may express as $$M=\frac{n}{V}$$ where $M$ now denotes molarity (not molar mass), and $n$, $V$ denote, as usual, number of moles and volume.
One thing to always keep in mind when calculating molarity, is that the volume, $V$, in the equation above needs to account for volume of the entire solution, not just the solvent. What this means is that if we are calculating the molarity of a solution of saltwater, we can generally just use the volume of the water, since the salt changes this value by a negligible amount. That said, if we are determining the concentration of ethanol (a liquid) in water, we need to account for the volume of both liquids and sum them to obtain our $V$.
Dilution Calculations
Moving on, the final thing we need to note is that we can use concepts of matter preservation to determine how concentrations change over time. For instance, let's say we know we have $3$ moles of salt in a 1L saltwater solution, and we decide to evaporate off half of the water in the solution. We know that the amount of salt in the saltwater hasn't changed, so we still have $3$ moles of salt, but we now have half the total volume of the solution. Surely this means we should have double the concentration? Indeed this is the case, and we can express this mathematically below. $$ n_{initial} = n_{final}, \quad \text{thus} $$ $$ M_{initial} V_{initial} = M_{final} V_{final} $$ This equation is often used to determine molarities in dilution calculations, where the initial and final volumes of a solution are known. This formula, and the one preceding it, form most of the theory upon which chemists rely, when working with concentration in laboratories.
Try out the molarity calculator and concentration converter to investigate how molarity is calculated, and how it may be converted to different representations of concentration.