The definition of a mineral is as follows: A naturally occuring, inorganic, crystalline solid with a fixed composition, or a composition that varies within a fixed range. So let's take each characterisitic on it's own, with some examples and non-examples, to clarify the the concept of "mineral."
naturally occuring: We as a species and a culture have learned to make many materials that do not occur naturally. Two that come to mind are silicon and aluminum. Now of course those elements occur naturally, but in nature, always bound to and combined with oxygen or other elements- never in a pure form on their own. So while the silicon in the chip on the computer I'm working on meets all the other criteria for "mineral," it dosn't occur in nature. Likewise with the aluminum in a can.
inorganic: This can be interpreted a couple of different ways, but my preference is that it is not composed of once-living material. The other is that it was not created by living organisms. Coal is clearly a rock, but it is composed of the remains of plants, thus it is not made of minerals. Again, the carbon atoms in coal have very likely spent time in mineral form (as carbonates), but the material as it occurs is derived from living matter. How about a fossil shell? This is why I prefer the former interpretation: the shell was never alive, so it is composed of minerals. Under the second interpretation, the shell was created by a living organism, and the crytalline material of the shell wouldn't be considered as minerals.
crystalline: The atoms making up the material must be organized in a predictable, symmetrical arrangement. Obviously we can't directly see how the atoms are organized, but the physical properties of the material, such as the form into which it grows, the way it breaks, the way it interacts with light and many others all reflect an internal order to the material's smallest units. In principle, if you know the position of a few of the atoms in a crystal, you should be able to predict the positions of all the other atoms. If you've had a little chemistry, it actually becomes a little eerie to consider just many repeating units can line up one after the other, almost flawlessly. This is the first criterion by which obsidian fails to meet "mineralness." Its atoms, while well-bound to each other, are basically random in their organization: knowing the position of any number of them will not allow you to predict the positions of any others.
solid: To simplify matters, I will simply define this as an inability to flow in response to the types of pressures (stresses) that occur in our day-to-day world. Almost anything will deform under the kinds of conditions that exist deep in the earth, so let's stick to the surface. Glass is sometimes described as a super-cooled liquid, and there is a widespread belief (though fallacious) that glass will flow simply due to the force of gravity if given enough time. For all practical purposes (outside of thermodynamics) it's fair to consider natural glass a solid. Incidentally, the way I've described the common forms of matter to elementary kids and teachers that seems to be effective (in that it "sticks") is solids don't flow; liquids flow, but don't compress very easily, and gasses flow and compress easily.
fixed composition: Some minerals have compositions that are fixed to within parts per million- that is, they may have some contaminants, but most atoms/ions that don't belong are excluded during the process of crystallization. Quartz SiO2 is a good example of this. Feldspars are a group of minerals that can vary a fair amount, but each type (plagioclase, for example) has a fixed range within which it varies. Plagioclase is particularly important, because the ratio of calcium and sodium tells a geologist much about the source and environment from which the mineral (and host rock) arose. So plagioclase is further broken down into 5 "subsidiary" minerals defined on which part of the compositional spectrum they fall into. Labradorite, which I have mentioned before, has a 50 to 70% calcium composition, with the remaining 50 to 30% of those spots occupied by sodium. Natural glass fails the mineral test in this respect as well; there are as many compositions of glass as there are of volcanic rock- which is a lot.
OK, that's a long-winded intro, but I expect I can refer back to it from time to time. As I indicated in the first paragraph, to a beginner this seems like a cumbersome defintion, but after you've applied it for a while, it rolls off the tongue as easily as anything.
I don't have most of my obsidian anymore, but I do have a few nice pieces. All of the following are high silica in composition, or in geo-speak, rhyolite glass. There was a period during which I got into knapping in a big way, so I had a bucket of the stuff to play with. Below is a flake that never got beyond the blank stage.
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