The short answer: Officially, no, Pluto is not a planet. In 2006, the International Astronomical Union (IAU) categorized Pluto under the new category of "dwarf planet," along with a handful of other small worlds throughout the solar system.
Why Pluto isn't considered a planet
According to the IAU, for an object to be a planet it needs to meet three criteria: It has to orbit the sun, be large enough that the force of its gravity pulls it into a spherical shape, and have "cleared the neighborhood around its orbit." In theory, an object that's large enough will clear its orbit either by collecting other debris with its gravity or by surviving impacts with it.
Pluto meets the first two requirements, but not the third, because it's part of a cloud of similarly sized debris that orbits the sun beyond Neptune: the Kuiper belt.
Because astronomers discovered Pluto decades before any of the other Kuiper belt objects (KBOs), they long assumed it was one of a kind and called it a planet.
But since 1992, they've discovered more than a thousand KBOs, including Eris, a dwarf planet that's just a bit smaller in diameter than Pluto and about 27 percent more massive. Rather than add all these to the list of planets, the IAU decided to create the new category of dwarf planet.
Unofficially, though, size is a part of the case against Pluto as well. Pluto, Eris, and the other dwarf planets are half as big as the planets in diameter and 10 times smaller in mass.
The whole idea of a "planet" is, admittedly, an arbitrary category, but if you have to draw the line somewhere, this seems to be a reasonable place to do it.
This isn't the first time we've demoted a planet
There's also some historical precedent for kicking objects off the planet list. Over time, the list of planets has varied widely, and until the mid-19th century newly discovered asteroids like Vesta and Pallas made the cut.
Eventually, though, astronomers realized those asteroids were among hundreds of others clustered in a belt. Rather than classify them all as planets, they were eliminated from the list altogether. In retrospect, Pluto's 1930 discovery and 2006 demotion has followed the exact same pattern.
The case for making Pluto a planet again
The IAU's Pluto decision quickly proved to be its most controversial one ever, and there are lots of scientists and space fans who'd like to see it overturned, for a few different reasons.
Their main objection is that the IAU's "clearing the neighborhood" definition is vague and illogical. There are no specified numbers for the size of a "neighborhood" or the number of other objects that can be there for it to still be considered "cleared." And the definition is fundamentally about an object's location, rather than its intrinsic qualities — if you moved Pluto to a different spot, in theory, it could suddenly become a planet.
Pluto supporters also argue that its small size shouldn't disqualify it from being a planet. Sure, it's an order of magnitude less massive than the inner rocky planets (Mercury, Venus, Earth, and Mars), but these in turn are an order of magnitude less massive than the gas giants (Jupiter, Saturn, Neptune, and Uranus). We could just as easily draw the line to include Pluto, Eris, and the other dwarf planets or draw it to exclude the inner rocky planets including Earth.
Finally, they note that the vast majority of Kuiper belt objects are irregularly shaped chunks of rock, whereas the four dwarf planets are rounded, satisfying the IAU's second criteria. Pluto also has moons and a thin atmosphere.
All these qualities mean Pluto and the dwarf planets have much more in common with the planets than most Kuiper belt objects. Instead of leaving them out, supporters say, let's add them all to the list of planets and allow it to grow with new discoveries in the future.
So is Pluto ever going to be a planet again?
For better or worse, there's no indication that the IAU plans to pick this topic back up anytime soon. The original reclassification stirred up a ton of controversy that continues to this day, and the astronomers who support the original decision have no wish to stoke it further.
During its brief flyby Tuesday, New Horizons revealed Pluto to be a fascinating, mysterious world. That's true regardless of however the IAU decides to categorize Pluto. But for the foreseeable future, it seems that astronomers will be considering it a dwarf planet.