Hole flow is the same thing as electric current, which is in the opposite direction of the actual flow of electrons. The term hole flow is usually only applied to semiconductor physics.

Around 1752, Benjamin Franklin developed his theory on the flow of electricity. Franklin believed that electricity flows like a fluid, and this fluid flows from areas of positive charge to areas of negative charge. It would be over 100 years before it was understood that current flow was actually the movement of charged particles.

By the time science understood that electric current was the movement of negatively charged electrons, it was too late to change the standards, the textbooks, the schematic diagrams, and the generally accepted theory. The direction of current flow was set as opposite to the actual flow of the charge carriers, which we now know flow from areas of negative charge to areas of positive charge.

In most cases, this is little more than a historical curiosity. In basic circuit analysis, it doesn't really matter that everything is backwards, because the standards and models still work. The behavior of the circuit is predictable. It doesn't really matter which direction the electrons are moving, because the concepts of voltage drop, electric fields, magnetism, and all the physical world that they interact with behave as we expect them to.

Electron flow direction, however, played an important part in the invention of the vacuum tube. It was important to understand that the negative charge carriers were the ones that were moving in order to understand the Edison effect (the boiling off of electrons) which formed the basis for the device, as well as to develop the control grid which makes a vacuum tube act as an amplifier.

The term "hole flow" came into use after the transistor was invented. A semiconductor depends on "holes" in the crystal lattice of the transistor caused by doping the semiconductor material. These holes are empty areas that the electrons can flow through in order to allow the material to act as a conductor rather than an insulator. In a p-n junction, which in its most simple case forms a diode, electrons can only freely travel from the p-type semiconductor layer (the side with the holes) to the n-type semiconductor layer. In the other direction (that is, n-type to p-type), the electrons do not see holes to enter the semiconductor material and the diode acts as an insulator. (This is a simplified explanation, but accurate enough for the purpose of this writeup. Click the hardlinks for more detail.)

From this, we can see that the difference between current flow and electron flow makes a big difference in truly understanding the physics behind semiconductor operation. To maintain the positive to negative flow standard when talking about semiconductors, we need to talk in terms of hole flow. That is, rather than discuss the direction the electrons are traveling (which would be the opposite direction of the electric current), we discuss the direction the holes are "traveling". This is very counter-intuitive, as empty space obviously does not actually move, but necessary to maintain the standards set by Benjamin Franklin back in 1752.