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Static electricity ionizes air molecules in clouds/upper atmosphere

Tiny pockets, or strings, or tethers of the ionized air floats down to the earth invisibly in all kinds of random lighter than air floating patterns, and when it eventually touches the ground it creates a temporary circuit and the ionized static energy in the clouds transfers to the ground and you can see the weird random pattern the ionized air molecules drifted to the ground in, in the shape of the lighting bolt. That’s why lighting bolts all look different and completely random weird shapes.

Think if you dropped a feather or leaf from high up and traced its pattern downward. It would be random and flowy and chaotic every time. Same with the ionized air that creates lightning, flowing to the ground.

Lots of thin invisible ionized air tendrils pretty much hanging from clouds and the upper atmosphere, and when one of them manages to touch the ground a transfer of energy can occur along that least resistant ionized trail

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We don't have a full understanding of all the processes involved in lightning formation, but the basics are that a large voltage or potential difference forms between cloud to cloud or ground. At a certain voltage, gases will enter a phase called electrical breakdown, which for air at atmospheric pressure is around 30 kV/cm. This is where the fun begins.

When electrical breakdown occurs in a gas, a chain reaction can form. Free electrons will be accelerated by the large voltage (I use voltage and electric field interchangeably from here on out, apologies if that's confusing, i was too far in and im too lazy to go back and edit) and will eventually collide with another particle in the gas. If the electron gains enough kinetic by the time it collides with that particle, it can ionise it, releasing a 2nd electron, which along with the initial first electron will be accelerated by the electric field. This process will repeat itself over and over again causing the number of free electrons and ions to grow exponentially, forming a conductive channel that propagates through space. This model works well enough at low pressures that you might find in a vacuum chamber but it's not so great at higher pressure like 1 atm. Why? Well because at low pressure there isn't a whole load of particles, so there isn't a whole load of collisions between particles, and therefore there isn't a whole load of energy transfer. This isn't the case at atm.

The next thing to add to the model is the large relative mass difference between electrons and every other particle in a gas. An electron is about 2000 times lighter than a neutron or proton, and as a result will gain a lot more velocity than say an ion of oxygen or nitrogen for the same electric field. This means that a localised charge distribution can form.

(Cavaet: This is where the understanding of lightning formation and propagation sort of becomes less understood and there are several theories. I'm going to stick with the one I'm more familiar with through my own work and research)

The charge distribution looks like a finger and depending on the orientation of the electric field the tip will be electrons or ions. In most lightning cases, it'll be electrons leading, as ground will be positively charged relative to the cloud. Here's a link that hopefully shows it Streamer discharge

In front of this charge distribution, more localised ionisations will build up pulling the streamer forward, towards ground, in a similar way to the avalanche breakdown, although the exact mechanism has several theories such as x rays, electron collisions and strong local electric fields. In all likelihood it's a combination of all three, plus several other factors.

The streamer will continue to propagate until it gets close to ground. At which point an even larger electric field can form due to the combination of negative electrons in the streamer and positive charge at the ground, with the field strength being the largest at higher points. If the new large field is strong enough, a secondary positive upward strike can also form. The upward strike is generally a lot hotter and more powerful than the downward strike, due to ions having a lot more kinetic energy than electrons. It's been a while since I've done any work in this area.

This is why you never want to stand under a tree or in open space and get as low as possible during a lightning strike.

So this streamer model that I've described has a few flaws. The most obvious one being that streamers only really work over short distances, around a few centimetres. Other models such as the leader model exist which try to explain the more large scale properties of lightning. It doesn't take into account the distribution of gas particles in the atmosphere, we know there are various layers etc. and has no chemical input. Plasmas like lightning will generate different oxygen, nitrogen and hydroxyl particles, like ozone or OH. In the past, I did a bit of theoretical work on what sort of impact OH would have.

A recent study that came out around Christmas proposed an oxygen meta state that could theoretically allow charge transfer but I haven't had a chance to fully look over it.

A complete picture of lightning would be either a simple model like the streamer model, or a very complicated physio-electro-chemical model. I'm happy with the streamer model for now :)