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I'm just having some difficulty with completely understanding the concept of static electricity. What actually happens? Is it because the electrons cant move? And if so, why can't they move?

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"Static electricity" is a rather poor name for the situation in which a large object possesses a net electric charge. "Static", with nothing after it, is a bad name for "excess charge". All objects made of ordinary matter(*) contain both positive and negative electric charges, of the same size individually, in the form of respectively protons and electrons. In most cases, there are equal numbers of these and thus the object possesses no net charge. In some cases, however, there can be an excess or deficiency of one with respect to the other, i.e. we have more electrons than protons, or more protons than electrons, in the object, and we say in that situation that a net charge exists because the sum of the two is then no longer zero. The reason we don't usually see objects with net charges is because the strong forces between charges tend to result in them finding a way to even things out - e.g. even once you manage to create an unbalanced charge on an object, when it's in atmosphere, it will tend to bleed or steal electrons (since protons cannot easily move) into or from the surrounding air until it's once again charge neutral.

"Electrostatics" is the special case of electromagnetic theory where we consider a physical situation in which electric charges are present but are not moving . A stationary physical object holding onto a net charge of "static electricity", that is not changing, is an example of the kind of situation dealt with in electrostatics. Mathematically, it consists of studying the problem of calculating the electric field $\mathbf{E}$ in the restricted case of the two Maxwell equations of Gauss's electric law and Faraday's law given respectively by

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\ \ \ \ \ \mathrm{and}\ \ \ \ \ \nabla \times \mathbf{E} = \mathbf{0}$$

where $\rho$ is the charge distribution in space - that is, the density of charge at each point therein, and such that this distribution does not change with time. The first equation says the sources of electric field are the charge distribution only, and the second says that the field conserves energy, i.e. that if you move another charge in a closed path around that field, you will end up with no new energy and no loss of energy after a full circuit, or to put it another way that may be a useful warning to any curious amateur who might be coming upon this post so as not to go down the wrong track, that electrostatic perpetual motion machines are impossible.

"Electrostatics" is a valid term, and acceptable to use. One should avoid talking about "static electricity", and it is high time we kick this word out of use. Instead, talk about "electrically charged objects" instead. Instead of talking of a build-up of "static", talk about a build-up of "charge". Do not talk about "anti-static" devices to clip to your hands when working around sensitive electrical devices, talk about "charge shunts" instead.

(*) That is, to exclude the still poorly-understood dark matter, of which if we know at least anything, likely contains no charges, that is, it is made of neutral particles only, otherwise it would not be "dark", i.e. invisible to light, where "light" here means all forms of electromagnetic radiation, both visible and invisible to the human eye.

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Static electricity is the term used when the charge doesn't move but stays where it is.

  • If you have rubbed your shoes over a carpet you might have left charge on the carpet. It stays there and is static until it is given a path to move through (when you touch it)

  • If you are charging a capacitor consisting of two plates, then soon those two plates are filled with oppositely signed charge. Disconnect the plantes from their surroundings but keep them close. This will now store the charge and it is static.

  • A battery might be the best example. Charge built up on the negative terminal/pole moves when given the possibility (when a path is created to the other terminal). Until then it is static.

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I am surprised that you think that the Wikipedia article does not answer your questions.

The answer to your question is open-ended and could be coupled with a comparison with current electricity.
In times past the two were thought to be different phenomena and it took some time to show that batteries and electrostatic generators were to do with the same electric charges.

In simple terms in the demonstration laboratory static electricity is to do with large voltages (thousands of volts) and very small currents (picoamps) and current electricity is to do smaller voltages (volts) and larger current (amperes).
For example a gold leaf electroscope can be used as a voltmeter and a simple one will only start registering when the voltages are in excess of several hundred volts.
The voltage produced by a 1000 V power supply can be measured using a moving coil voltmeter through which a current flows but also with a gold leaf electroscope which only draws current until the charge on it reaches a steady value.

My mention of very small currents to do with electrostatics indicates that currents do flow until a system achieves an equilibrium state.
Static electricity is to do with charges which reside on insulators so once an excess of one type of charge is generated (during which time there will be a movement of charge) the charges do not move.
Here the is some ambiguity as to what is meant by the term insulator and for electrostatics this usually means that during the course of a period of time the charge does not flow away.
A glass rod can be charged by rubbing but if it has a surface layer of moisture the charges will flow off it as water is a reasonably good conductor. Wood is a good example of being a conductor when electrostatic experiments are because of the water which it has absorbed.

Again in broad terms electrostatics deals with small amount of charge as compared with current electricity but that generalisation can easily be shown to be limited eg lightning represents currents which are thousands of amps but they last for less than a second.

So the terms static electricity and current electricity are rather loose terms and depend on the circumstances just as do the terms conductor and insulator.
When I comb my hair the comb which is made of plastic (an "insulator") becomes charged and the charges stay on it for long enough for me to pick up small pieces of paper and I would call that an example of static electricity.

So as you go through the Wikipedia article on static electricity you need to think about how it is that the charges are, for a period of time, in some way unable to move.

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