Proofing is an interesting process, and for some reason it is called 'proofing', not 'proving'. I suppose it is a process of fermentation, but the production of alcohol isn't the aim.
Proofing is all about ripeness. A good loaf of bread comes from three or more good proofs, each one being ripened for long enough for the gas bubbles to fully expand inside the gluten bubbles which make up the dough. This expansion is paused by 'degassing'. This process is often referred to as 'punching down', though I have no idea why anyone would want to punch dough. Some weird testoserone fantasy, like dough was some uncontrollable beast that grows and grows, and must be defeated.
The three proofs are more like a progression towards a finished shape - each one combing the gluten network into a more harmonious and structurally sound configuration, able to withstand the sudden heat to come when the shaped dough is finally exposed to the hot oven without losing its form.
Most breadmaking books I've seen only talk about two proofs. My theory is that when bread is baking in tins, two proofs are enough, in most cases. When baking on the sole of the oven, there are no supports to hold the dough up as it kicks. If you look at each proof as a 'tightening up' of the gluten network, then the third proof simply gets everything nice and tight, ready for baking.
The first proof, in bakery jargon, is referred to as the bulk proof, or primary proof. It's the time when all the primary fermentation action is happening in the dough. Depending on temperature, the first proof should take about four hours if the dough is made and held at 24 degrees celsius. You can slow it down by lowering the temperature in the fridge - I like my bulk proof to go for at least 12 hours. This allows the fermentation process to grow big gas bubbles inside the dough. If you poke your finger into the ripened dough at any stage in the three proofing processes, it will go straight in. There will be no rubberiness or resistance, and the hole you make will hold for at least a few minutes before closing in.
The second proof, or cut proof (technically known as the 'intermediate' proof) is so named because it is the proof that is started when the dough is divided from the bulk dough. It assumes that you are making more than one loaf of bread from the dough. You will note that in this website, we pretty much always make about 1.6 kilos of dough, so that you get two 800 gram loaves. Of course, you can just make one big loaf out of this, so no cutting is involved - but I like to always make a loaf for a friend or neighbour, as part of the idea that the baker's job involves generosity.
Once the dough is cut, it has to be rounded, so that there is a skin around the gluten network. We call this 'rough rounding' in the bakery, as it is generally done fairly quickly, and the result is to seal the gas into the ball of dough so it doesn't bleed out during proofing. At the same time, the process of rough rounding squeezes and compresses the gas contained in the dough. Some bakers call this 'degassing', and to a certain extent this is what is actually happening - though at this stage our primary purpose is to 'comb' the gluten network into a ball for further fermentation to occur.
Rounding is not kneading. Think of it as sealing in the gas, or tucking it in. The base of the dough will have a seam, which rests on the underside, while the top should be one smooth skin. Thus, the gas can't escape.
The rounded balls of dough need to fill up with gas again. Each time this occurs, the windows of gluten grow thinner, their texture as a network becomes a bit more intricate as well. The structure of the whole is becoming stronger too, leading to a more pleasing shape in the final proof. You will notice that a well proofed dough, even when baked in a tin, rises straight up when it hits the oven, whereas one that has been taken before full ripeness is achieved will tend to go more sidewiays than up when baked - in a tine this is what I call a 'muffin top'. Sole baked bread will be more oval shaped rather than round.
The third and final proof, or moulded proof, involves taking the ball of fully ripened dough, which has been ripening now for at least 2 hours at about 24 degrees C, or overnight if left in the fridge, and squeezing it into a cylinder. There are numerous ways to do this - you can turn the dough upside down on the table, so that the seam side is upwards, and dock it down with your fingers to remove the gas, and then roll it up into a cylinder. Or, you can simply squeeze the ball into a cylinder shape, carefully maintaining surface tension on the clean (top) side of the dough. Once this is done, you simply dust it with flour or semolina, or not, as is your choice, then place it on a wooden board that's also covered thickly in Semolina, and let it grow in readiness for the bake.
I like to score the dough at this point, but you can also score it just before it goes in the oven. If you score it now, the score marks have time to open up nicely during the final proof, leading to a clean cylindrical shape when it's baked, without 'hills and valleys'. If you score just before placing in the oven, you will get little ears kicking up from the dough, which are currently very fashionable in the artisan bread world - though I find these ears somewhat impractical when it comes to putting the bread into a paper bag later on, as they tend to tear the bag. They also have a tendency to cut your fingers, as they can be quite sharp.
The final proof, using this method, is quite fast when the temperature is held to around 24 degrees C - only an hour or so. Of course, you can also cover the dough with a plastic box and place it in the fridge to bake tomorrow.
So, that's a quick guide to the three proofs. In a nutshell, make sure that each stage is nice and ripe before moving to the next. As a rule of thumb, ripe does not feel at all rubbery when you poke it with your finger. If it resists you, it isn't yet ripe. Patience is called for - or possibly a bit more temperature!
Happy Sourdogh Baking!