A much more thorough discussion of the classification of matter, and how it has changed since ancient times, can be found on this excellent page at the Chemogenesis site.

Since this latter schema covers virtually all matter that we encounter in the world, it is important that you thoroughly understand the meaning of each part of it. We will begin by looking at the distinction represented in the top line of the diagram.

Homogeneous matter (from the Greek homo = same) can be thought of as being uniform and continuous, whereas heterogeneous matter (hetero = different) implies non-uniformity and discontinuity.  To take this further, we first need to define "uniformity" in a more precise way, and this takes us to the concept of phases.

A phase is a region of matter that possesses uniform intensive properties throughout its volume.

A volume of water, a chunk of ice, a grain of sand, a piece of copper— each of these constitutes a single phase, and by the above definition, is said to be homogeneous.

A sample of matter can contain more than a single phase; a cool drink with ice floating in it consists of at least two phases, the liquid and the ice. If it is a carbonated beverage, you can probably see gas bubbles in it that make up a third phase.

← The phase boundaries in an oil-vinegar salad dressing are easy to see...

...but those in milk are so small that they are only revealed by scattered light. →

Getting back to our classification, we can say that

Homogeneous matter consists of a single phase throughout its volume; heterogeneous matter contains two or more phases.

The air around us, most of the liquids and solids we encounter, and all too much of the water we drink consists not of pure substances, but of mixtures. You probably have a general idea of what a mixture is, and how it differs from a pure substance; what is the scientific criterion for making this distinction?

To a chemist, a pure substance usually refers to a sample of matter that has a distinct set of properties that are common to all other samples of that substance. A good example would be ordinary salt, sodium chloride. No matter what its source (from a mine, evaporated from seawater, or made in the laboratory), all samples of this substance, once they have been purified, possess the same unique set of properties.

A mixture, in contrast, is composed of two or more substances, and it can exhibit a wide range of properties depending on the relative amounts of the components present in the mixture. For example, you can dissolve up to 357 g of salt in one litre of water at room temperature, making possible an infinite variety of "salt water" solutions. For each of these concentrations, properties such as the density, boiling and freezing points, and the vapor pressure of the resulting solution will be different.

Finally, we all prefer to drink "pure" water, but we don't usually concern ourselves with the dissolved atmospheric gases and ions such as Ca²⁺ and HCO₃^– that are present in most drinking waters. But these harmless "impurities" and are always present in those "pure" spring waters.

However, these and similar substances could seriously interfere with certain uses to which we put water in the laboratory, were we customarily use distilled or deionized water.

Even these still contain some dissolved gases and occasionally some silica, but their small amounts and relative inertness make these impurities insignificant for most purposes. When water of the highest obtainable purity is required for certain types of exacting measurements, it is commonly filtered, de-ionized, and triple-vacuum distilled. But even this "chemically pure" water is a mixture of isotopic species: there are two stable isotopes of both hydrogen (H¹ and H², often denoted by D) and oxygen (O¹⁶ and O¹⁸) which give rise to combinations such as H₂O¹⁸, HDO¹⁶, etc., all of which are readily identifiable in the infrared spectra of water vapor. (Interestingly, the ratio of O¹⁸/O¹⁶ in water varies enough from place to place that it is now possible to determine the source of a particular water sample with some precision.) And to top this off, the two hydrogen atoms in water contain protons whose magnetic moments can be parallel or antiparallel, giving rise to ortho- and para-water, respectively.

The bottom line: To a chemist, the term "pure" has meaning only in the context of a particular application or process.

Consider a heterogeneous mixture of salt water and sand. The sand can be separated from the salt water by the mechanical process of filtration.

These examples illustrate the general principle that

Heterogeneous matter may be separated into homogeneous matter by mechanical means.

Turning this around, we have an operational definition of heterogeneous matter: if, by some mechanical operation we can separate a sample of matter into two or more other kinds of matter, then our original sample was heterogeneous.

To find a similar operational definition for homogeneous mixtures, consider how we might separate the two components of a solution of salt water. The most obvious way would be to evaporate off the water, leaving the salt as a solid residue. Thus a homogeneous mixture can be separated into pure substances by undergoing appropriate partial changes of state— that is, by evaporation, freezing, etc.

Note the term partial in the above sentence; in the last example, we evaporate only the water, not the salt (which would be very difficult to do anyway!) The idea is that one component of the mixture is preferentially affected by the process we are carrying out. This principle will be emphasized in the following examples.

Some common methods of separating homogeneous mixtures into their components are outlined below.

Note that if all the liquid were boiled away, the distillate would be identical with the original liquid. But if, say, half of the liquid is distilled, the distillate would contain a larger fraction of the more volatile component.

If the distillate is then re-distilled, it can be further enriched in the low-boiling liquid.  By repeating this process many times (aided by the fractionating column above the boiling vessel), a high degree of separation can be achieved.

Similarly, a molten mixture of several components, when slowly cooled, will first yield crystals of the material having the highest melting point. This process occurs on a huge scale in nature when molten magma from the earth's mantle rises into the lithosphere and cools underground — a process that can take up to a million years.  This is how the common rock known as granite is formed.  Eventually these rocks rise and become exposed on the earth's surface.

Liquid-liquid extraction

In working with natural products such as plant materials, a first step is often to extract soluble substances from the plant parts.

This, and similar extractions of the soluble components of complex  solids, is carried out in an apparatus known as a Soxhlet extractor.

The idea is to continuously percolate an appropriate hot solvent through the material, which is contained in a porous paper "thimble". Hot vapor from the boiling flask bypasses the extraction chamber through the arm at the left (labeled "vapor" in the illustration →) and into the condenser, from which it drips  down into the extraction chamber, where a portion of the soluble material mixes with the solvent.  When the condensate reaches the top of the chamber, it flows out through the siphon arm, emptying its contents into the boiling flask, which becomes increasingly concentrated in the extracted material.

The advantage of this arrangement is that the percolation-and-extraction process can be repeated indefinitely (usually hours to days) without much attention.

[Image sources: left, Wikimedia; right, unknown]

This Wikipedia page shows a nice animation of the process.

In this video (4 min), the capsaicinoids ("hot stuff") in peppers are separated and concentrated using a Soxhlet extractor.

As a liquid or gaseous mixture flows along a column containing an adsorbant material, the more strongly-adsorbed components tend to move more slowly and emerge later than the less-strongly adsorbed components.

In this example, an extract made from plant leaves is separated into its principle  components: carotene, xanthophyll, and chlorophylls A and B.

Although chromatography originated in the mid-19th Century, it was not widely employed until the 1950's.  Since that time, it has encompassed a huge variety of techniques and is no longer limited to colored substances.  Chromatography is now one of the most widely-employed methods for the analysis and separation of complex mixtures of liquids and gases. More: Introductory survey; Wikipedia page

Since chemistry is partly the study of the transformations that matter can undergo, we can also assign to any substance a set of chemical properties that express the various changes of composition the substance is known to undergo. Chemical properties also include the conditions of temperature, etc., required to bring about the change, and the amount of energy released or absorbed as the change takes place.

The properties that we described above are traditionally known as physical properties, and are to be distinguished from chemical properties that usually refer to changes in composition that a substance can undergo. For example, we can state some of the more distinctive physical and chemical properties of the element sodium: