Confusion on Concentration

Concentration, in the scientific and technical field, is the amount or fraction of a given substance in a mixture. In some contexts it can be imagined as the "crowding" of molecules in a given space - more on this later.

Knowing concentrations has a huge practical importance. It may make a difference between an ore body that is worth mining and one that isn't; the difference between a medicine that cures and one that doesn't (or even kills) and so on.

It also can make the difference between a reaction that proceeds smoothly, and one that suffers an explosive runaway. This is because reaction rates are proportional to (some power of) the collision frequency between reactant molecules, which is in turn proportional to (a function) of concentration of reactants - their crowding in a region of space.

There are many ways to express concentration, each tailored to specific needs - and sometimes, to achieve a specific effect. For example, it's more scary to say that the atmospheric concentration of carbon dioxide is close to 400 ppm (by volume) than to say it's close to 0.04% (by volume).
Concentration can also be expressed on different bases - mass, volume, moles etc - it's important to know clearly which base is being used to avoid mistakes.
While it's often practical to express concentrations on a volume basis, there are two conflicting issues. While mass is absolute, volume is a function of temperature, and this introduces a further complication; on the other hand, it's easier to handle liquids than solids.

A very common expression for concentration, used extensively in industry and medicine and also chemistry is percent (%). Percent is straightforward: it says how many units of a particular substance there are in 100 units of mixture. A typical application works like this: in order to prepare 1o liters of 5% erbicide solution you measure half a liter of erbicide formulation and dissolve it in 9 and half liters of water; great accuracy is not required.
Usually percent is expressed on a weight basis, but it can be by volume or mole - for mixtures of gases at low pressure, percent by volume and mole are identical. For dilute aqueous solutions only, water can be measured by volume rather than weight. UPDATE: I made some confusion: the density of dilute aqueous solutions can be taken as 1 - while this is erroneous for concentrate ones.

An unit that is often encountered while discussing pollution and environmental issues is the part per million (ppm). It means that there is one part of a chemical in one million parts of mixture. Usually it is on a volume or mass basis - one gram in a metric tone is one ppm. The ppm is intended to be used for low concentrations, and in my humble opinion using numbers like 100 000 ppm is misleading.

Chemists like to think in terms of moles. That's because chemical reactions are written in terms of moles (which are defined as a fixed number of molecules), so it makes sense to express also concentration as moles on volume or mass. So the former is molarity (moles per liter of solution) and the latter is molality (moles per kilogram of solution). Molarity is used very often, also because it fits well to volumetric analysis: when the volume of a titrating solution of known concentration is known, working out the number of moles is straightforward. For example, 10 mL of hydrochloric acid 1M contain 0.01 moles of HCl. In the case of pure substances, we talk of molar density rather than molar concentration.
Molality doesn't enjoy widespread use, mainly because it becomes useful only when discussing rather subtle things such as activity of solutes: dealing with activities is more complex and is required only for precision work or in particular fields.

Another expression for concentration (that I personally like a lot) is the mole fraction (χ), defined as the number of moles of one component divided by the summation of the number of moles of all components in the mixture. It has some interesting properties: the partial pressure of a gas in misture is equal to the product of mole fraction by total pressure, and it happens that equilibrium constants and often reaction rates for gas-phase reactions are expressed as a function of partial pressures. Some properties of solutions are proportional to the mole fraction of solute or solvent, and mole fractions are the coefficients to use in linear combinations - for example, if a gaseous mixture has to be treated as a single gas, its properties are given by a linear combination of the properties of the different components.

All concentration units can be converted into another with some maths. However, it's a tricky practice and often it requires to know the density of the solution or mixture. But density is rather troublesome quantity: difficult to predict even for simple systems, it's also difficult to measure with good accuracy.