Have you ever played the sliding-15 puzzle? There are fifteen sliding pieces, numbered 1 to 15, arranged in a 4x4 grid. It has one "hole" where there is no piece. You can move any piece adjacent to the hole into that space.
As you play the game, what is moving: the numbered pieces, or the hole?
Technically speaking, you are moving the numbered pieces. They are the electrons, the physical objects that move around. You can even get them to move in a complete path (a closed circuit).
But if you have played this game for enough time, you quickly realize how important the hole is to the strategy of the game. A skilled player can move the hole anywhere he wants, and the hole can even move around in a closed circuit. The hole represents the absence of electrons, and can move around just as the electrons themselves can. In semiconductor theory, we even officially call them "holes" and treat them like a particle, even though they're not.
Whichever way you count, there is something moving. In electricity, we call this "charge". Ben Franklin called it "electric fire" (see below).
Notice that whatever move you make, there are always 15 numbered pieces and 1 hole. We say that this is conservation of charge, and was discovered by Franklin (see below). As long as we are counting or adding things, it makes sense to make one kind of charge positive and another kind negative. How shall we assign which is which?
The problem is that there has never been -- and never will be -- a device which can see electrons or holes. They're just too small and too fast.
As @endolith furthermore points out, there are many things other than electrons that can flow to produce electric current. Nonetheless, physicists like to choose conventions.
Franklin asked several of his English colleagues for books, to establish a scientific library in America. In early 1747, the English naturalist Peter Collinson sent Franklin a glass tube along with a shipment of books, with directions on how it was being used in electric experiments. Apparently, physicists thought it was great fun to rub the tube and give each other shocks. How fun!
On May 25 1747, Franklin wrote a letter back to Collinson, first thanking him for the gift, describing some of the experiments he had performed with it, and then explaining his conclusions.
Franklin described his construction of a simple electroscope, electrifying it with the tube, and then de-electrifying it with a nearby needle. Franklin concluded that there is some kind of "element" that is being "drawn off or thrown off". He also noticed a glow when these experiments are performed in the dark, "like that of a Fire-Fly or Glow-Worm". Franklin therefore called it "electric fire", which we now call charge. He made a case that this is a new chemical element (to join earth, water, air, and fire).
He then described several other experiments, but his interpretation of the following experiment is the most important:
We suppose as aforesaid, That Electrical Fire is a common Element, of which every one of the three Persons abovementioned has his equal Share before any Operation is begun with the Tube. A who stands on Wax, and rubs the Tube, collects the Electrical Fire from himself into the Glass; and his Communication with the common Stock being cut off by the Wax, his Body is not again immediately supply’d. B, who stands upon Wax likewise, passing his Knuckle along near the Tube, receives the Fire which was collected by the Glass from A; and his Communication with the common Stock being likewise cutt off, he retains the additional Quantity received. to C, standing on the Floor, both appear to be electrised; for he having only the middle Quantity of Electrical Fire receives a Spark on approaching B, who has an over-quantity, but gives one to A, who has an under-quantity.
If A and B touch each other, the Spark between them is stronger, because the Difference between them is greater. After such Touch, there is no Spark between either of them and C; because the Electrical Fire in all is reduced to the original Equality. If they touch while Electrising, the Equality is never destroyed, the Fire only circulating.
Hence have arisen some new Terms among us. We say B (and other Bodies alike circumstanced) are electrised positively; A negatively: Or rather B is electrised plus and A minus. And we daily in our Experiments electrise Bodies plus or minus as we think proper. These Terms we may use till your Philosophers give us better.
Benjamin Franklin, Letter to Peter Collinson, May 25, 1747. Emphasis added.
Franklin had discovered the conservation of charge. In particular, he claimed that his "electric fire" only circulated, was never destroyed, and remained in equality of amount. As the electric fire came from the glass rod, it made sense to him that objects gaining electric fire were positively charged.
Glass is one of the most triboelectrically-positive substances; only polyurethane (not yet invented) and human skin and hair are more so. Had Collinson instead sent Franklin a rod of sulfur -- which is triboelectrically-negative -- it is likely that we would be talking about positively-charged electrons today. I find it poetic that "electric fire" can be made from "brimstone".
Franklin was somewhat right about "These Terms we may use till your Philosophers give us better." Indeed, it was an Englishman (physicist J. J. Thomson) who discovered that the actual particles moving in a circuit -- which he named "electrons" -- are negatively-charged. Unfortunately, even though something "better" came along, the convention stuck.
Returning to the original question, why is the convention "wrong"? The most common charge carrier (the electron) flows in one direction, but the direction of charge current is expressed in the opposite direction. The opposing directions create confusion among learners: it is not intuitive, it is confusing, learners often apply the wrong direction (or ignore direction), and it frustrates learners. When you are learning electricity, it is thus "wrong".
There are times when it is better to use electrons, and other times when you should use current. Just as the sliding puzzle has times when you want to move a numbered piece, and other times you want to move a hole. A skilled practitioner knows which one to apply, and doesn't even think about the difference.