As Kyle Kanos stated, the convention is largely due to Ben Franklin's work on electricity, and his theory of a "single-fluid model".
Particle physicists (such as they were in the 1700s) followed the "corpuscularian" atomic model that we call the "plum pudding" model today; the particles that formed matter were spaced relatively evenly apart and evenly distributed, forming a loose "fluid", and different types of matter had different types of particles and at different densities, which gave them their properties such as weight, phase, malleability, and yes, electrical conductivity. Most people who worked with electricity thought that different types of particles carried opposing charges, and electrical current thus involved a "two-fluid transfer"; positively-charged particles moved from surplus to deficit, and same with negatively-charged particles, creating an equilibrium.
Ol' Ben thought a little differently; he saw a propagation of current from only one end of a connection between charges, through experiments conducted with a Leyden jar to store a static charge gathered either with friction or from lightning, and then discharging it through matter with different electrical resistance, including, as the stories go, his party guests. The people at the far end of a chain of people holding hands reacted last (and least) to the discharge from a Leyden jar, instead of those in the middle as would be expected from the prevalent two-fluid model. So, he proposed that while oppositely-charged potentials did seek to equalize, only one "charge carrier" was an actual moving "fluid" in this circumstance, and the other potential was simply caused by a deficit of this fluid charge carrier, creating a relative surplus of a "fixed" charge carrier distributed evenly through the material.
It was a genius proposal at the time. There was only one problem; he could not devise an experiment or instrument that could detect which of the opposing charges, positive or negative, was being transported fluidly by the charge carrier. He had to guess, and as luck would have it he guessed wrong; he documented the fluid charge carrier as being "positive", describing the charge as flowing from positive to negative. Since lightning is often observed traveling from the clouds to the ground, the negative pole of a DC circuit came to be called the "ground side" for this reason.
In the following years, Hans Christian Oersted discovered, purely by accident while demonstrating resistive heating of a metal coil, that the wire through which a current passed caused the needle of a nearby compass to deflect from true north, thereby demonstrating electromagnetism. His work was duplicated, leading to experiments and mathematical models predicting the force vectors of magnetic fields based on the direction and strength of current in the wire (Biot-Savart Law, Maxwell's Equations and the Lorentz Force Law). Around the same time, Sir William Crookes was experimenting with vacuum tubes, passing a strong electrical charge through them, causing the glass of the tube (used as the insulator and for visibility) to phosphoresce.
It wasn't until 1897 that J.J. Thomson, while experimenting with these "cathode ray tubes", connected the dots; using a thin cross-shaped sheet of mica placed within the tube, he showed, based on the "shadow" the cross forms on the wall of the tube, that what is passing through the tube is some sort of particle, which is being reflected by the mica. He then showed that these particles had to be negatively charged, because they were reflected by the mica sheet on the side of the negative pole, and were affected by the magnetic field of a permanent magnet as a negatively-charged particle would be, in accordance with the Biot-Savart Law and Maxwell's Equations. He reasoned that this negative charge carrier must be of lower mass than any other particle that makes up matter, otherwise some other particle would be moving to carry the charge (creating a more detectable change in mass; in fact this difference is detectable, but the ratio between charge carrier masses is over 1800:1). He named this particle the "electron" and asserted that it, and not any positive charge carrier, was most directly responsible for electromagnetism.
However, it was far too late. The convention that current flows from the positive to the negative of charged dipoles had been in common use for almost 150 years, and a lot of the work that ended up disproving it was, ironically, documented using it. Nowadays, we recognize that the movement of electrons is from the negative charge to the positive, but we diagram the movement of current in the opposite direction, as the propagation of a "positive charge", even though we now know better. That's why the positive lead or terminal is the red one, even though the source of the electrical charge is actually the negative "ground", while the actual ground in a lightning strike has a relative positive potential caused by the movement of air over it.
It's only with alternating current that we regain some sanity, because in practical terms no one wire is "positive" or "negative"; the black or red wire (in U.S. home wiring codes) is the "hot" side, on which the potential change is being actively driven by the generator; the white wire is the "neutral" or undriven side; and bare (or green) is a safety "ground"; U.S. codes typically state that the ground wire goes to the same terminator on the service panel as the neutral instead of directly to the actual earth, but either way it provides an easier path for a short circuit than through a person.