Short answer: Pluto has only two of the three necessary characteristics to be called a planet. Pluto has not cleared its neighborhood, or orbit. But, of course, there are additional details.
The simplest reason that Pluto is not a planet is that planet experts say so, and this is their job. But you may be looking for a more detailed explanation.
Let’s look at what defines a planet. This could be a very long and tedious discussion, because “planet” is an ancient concept used long before scientists knew very much about them. Also, frankly, in many areas of science the definition of a thing, perhaps counter-intuitively to non-scientists, is often pretty irrelevant to its study. Definitions that change over time that are never quite in line with the phenomenon being observed, etc. may seem like an impediment to science, but they often are not. The definition of a “gene” has changed dramatically as we’ve learned more about them, but this shifting description has not hampered genetic research. To some extent this may be the same with planets. A “planetologist” who studied Pluto back when it was still counted as a planet would not have to find a new job when our solar system went from 9 to 8.
The International Astronomical Union has settled on a set of definitions of solar system bodies, which includes planets, dwarf planets (which are mostly minor planets), small solar system bodies, trans-Neptunian objects which also might be called plutoids (those are also minor planets) and some small solar system bodies (including some comets) and satellites, and satellites are, of course, things that go around things that are not the Sun. Confused? Probably, but that is not a big problem because while these various identified flying objects have complex overlapping categorical status, one type of object does not. Planets are planets and they are not anything other than planets.
To be a planet, you have to be in orbit around the Sun. This would rule out the Moon, which, if it was in orbit around the Sun instead of the Earth, could well be a planet.
To be a planet the object has to have sufficient mass to have been shaped by gravity to be (mostly) a globe. This depends on various things so at the low end of the mass spectrum there will be different masses and different sizes of things that don’t quite make it to globular status.
To be a planet the object has to have cleared its orbit. In other words, as an object orbits around the sun, it is likely to bump into other objects. Over a period of time, the object has finished bumping into everything it is likely to bump into, and thereafter has only a low probability of bumping into something. That does not rule out something bumping into the object, of course.
A globe shaped object that goes around the sun but that has not cleared its orbit is classified as a “dwarf planet.” This is of course historically contingent. In the early days of a solar system, perhaps there would be large star-circleing round things that have not yet cleared their orbit. This speaks to the strangeness of definitions alluded to above. The definition we use today to classify our solar system’s objects applies to a solar system developed to the extent ours is developed. The IAU nomenclature would probably need revisions if applied to all planetary-star systems in the Universe.
This scheme is not without its critics and there is indeed debate. Some of that debate is a bit nitpicky but still interesting. For example, Alan Stern, with NASA, notes that many planets have not really cleared their orbit, noting in relation to the Pluto controversy, “If Neptune had cleared its zone, Pluto wouldn’t be there.” Yes, apparently heavenly bodies have irony.
Anyway, as implied, Pluto is not classed as a planet because it has not cleared its orbit. Therefore it is a Dwarf Planet. Since it is far away (farther than Neptune) it also gets classed as a Trans-Neptunian Object. Furthermore, it is a Plutoid. That is simply a newer term applied to Trans-Neptuina dwarf planets.
The term Plutoid, then, refers to a dwarf planet, which for various reasons is apparently always specifically an ice dwarf, which is a trans-Neptunian body (orbiting most of the time beyond Neptune) that is sufficiently massive to be shaped like a globe. This term, plutoid, is officially adopted A plutoid or ice dwarf is a trans-Neptunian dwarf planet, i.e. a body orbiting beyond Neptune that is large enough to be rounded in shape. The term plutoid was adopted by the International Astronomical Union’s Committee on Small Bodies Nomenclature, but not by the working group on Planetary System Nomenclature. So you can use Plutoid or Dwarf Planet, or Ice Dwarf, depending on whom you wish to annoy.
Pluto, Eris, Haumea, and Makemake are the only known Plutoids. They are small enough and far enough away that more could be discovered.