It's common knowledge that electrons flow from negative to positive, but I have noticed than often the direction of the current is ignored. For example, the resistor is often put AFTER the LED, or the diode is put the opposite way. Why is the direction of flow often disregarded in electronics?
Electrons have a negative charge. Current is Coulombs per second. Coulombs are positive, so a coulomb moving one direction is actually caused by electrons moving the other direction in meta.
When we discuss current we are discussing the flow of positive charge particles. If the flow of current is actually made up of negative particles flowing in the opposite direction it makes no difference, that is two negatives which cancel. It is just a case of math and sign convention.
The only time you pay attention to the actual carriers is in something like a semiconductor where you need to know what is happening as you travel from electron carriers in the conduction band "holes" in the carrier band. The holes are positive charge carriers, but that is because we are counting the absence of an electron, the actual current is still made up of many many electrons slowly drifting.
Is current always electrons?
Actually if you ever model electrical systems in the body you will find that you can accurately model a neuron using a transistor network and such. Much of the current relates to ions like potassium. This means that you really do have motion of positive charge articles. It still is just drawn as a schematic because it does not matter what the charge carrier is as long as your schematic models the electrical properties well.
Is the electron moving the power?
Often people think that your power you are sending is the electron. In reality you are sending electromagnetic signals. You can slow the speed of your signal(ie. the power) propagating down a long pair of wires(one the signal and one the signal return) by changing the dielectric between them. This means two unshielded copper wires just sitting in space will actually have their signal travel near the speed of light. Your coax cable will probably travel very close to two thirds the speed of light. The electrons drifting are a function of the electric field that is present. If you were to measure how fast the electrons are drifing you would find it on the order of a few meters per second.
As you noted, people often don't know and don't care. Fortunately, For 99% of people out there it doesn't matter. Convention is that it flows from + to -, and it is useful for all engineers to stick with that convention simply to make talking with other engineers easier.
The only people it really matters to are either people designing chips (not people designing WITH chips), and some physicists. Some people believe that it really is important, but they are usually the pedantic drunks at frat parties that nobody wants to be around.
For the record, the current limiting resistor often found next to an LED can go on either side of the LED with no ill effects.
The only reason why the current direction is disregarded in scenarios like that is because it doesn't really matter in those scenarios.
There's no current until the circuit is complete and the circuit is not complete until you connect both the LED and the resistor. Once those are connected in series it doesn't matter which follows which in the circuit since the purpose of the resistor is to limit the current in the circuit and the latter depends on the sum of resistances of the LED and the resistor (and other parameters which are not changed when you swap the resistor and the LED, so I just ignore them in this answer) and that sum doesn't depend on whether the resistor is after the LED or before it.
So yes, maybe ideologically it would feel better to connect them in a specific order so that current "doesn't reach the LED directly, but only through the resistor", but practically it makes no difference. And believe me, in cases when it makes a difference (like super high voltage which makes the insulation break down if resistance happens a bit lower than needed) noone disregards things that seem minor. And no, I don't have any idea if current direction matters much in high-voltage scenarios.
May I drop some pedantic theory from my university studies? :)
As the other guys pointed, the current that flows from a "plus" to a "minus" is only a conventional way to represent the phenomena. This is due to the fact that electrons have by definition a negative charge, and probably this fact itself is a convention that prefers to give a positive sign to the protons which are in the core of the atom. Then, dealing with negative values (that come out from negative charge blah blah blah) it's annoying, hence the decision to consider the current as opposite to the movement of electrons.
A story about potentials and fields
Another point of view is that, always because of the negative charge carrier, the electric potential (that define voltages) is negative where there are more electrons, so it's positive where the electrons are less, and you would expect that the current flows from the higher potential to the lower, as the objects when are falling.
This has no influence on the order of components in the same branch of a circuit, as the current (for the principle of conservative fields and blah blah blah) is the same in the whole branch. For a more deeper analysis see this. Consider it like a pipe with pressured water: doesn't matter (theoretically) if a turbine is before or after a bottleneck, as the latter will have in any case an influence to the amount of water that flows in the pipe.
The diode, is still simple to understand what it does (basically the current flows in one direction and not the other; opposite things for electrons) and more complicated to understand why it does this way.
And about the "holes" thing, they are used because in semiconductor physics, and more when working with doped semiconductors, there are materials (or better, doped materials) that have less electrons in the valence band, and take electrons from near places in the conduction band, creating the current. But this is far easier if talking about holes travelling in the conduction band