The short answer is bureaucracy.

While there might be the option to volunteer (or not) when a few men were needed for a small trench raid at night, or to help bring in a wounded man, or for some other small-scale enterprise, for larger actions your fate was largely decided by paperwork shuffled at the Brigade or Divisional level: if your unit was the one assigned to the first wave of an attack then you all went.

If Divisional staff decided that a strong point or terrain feature in their sector should be taken, for example, planning would gradually filter down the chain of military bureaucracy, with each layer making more specific decisions for the men and resources required to achieve that plan. The Brigade commander might decide which battalion(s) should attack, and what artillery and other assets to make available; then the battalion commander might decide which companies to put into the attack and in what order; and as it filters down to the company and platoon and section level less and less autonomy can be exercised: the plans have been made higher up, and it is your role to carry them out as prescribed. This lack of control was keenly felt amongst the front-line troops of all nationalities; this description was written by an Italian soldier:

This is the war. It is not the risk of dying, not the red firework display of a bursting shell that blinds us as it comes whizzing…but the feeling of being a puppet in the hands of an unknown puppeteer – and that feeling sometimes chills the heart as if death itself had taken hold of it. Chained to the trench until orders to be relieved arrive as suddenly as a cannon shot or a snowstorm, tied to ever-present danger, to a fate that is inscribed with the number of your platoon or the name of your trench, unable to take your shirt off when you want to, unable to write home when you want to, seeing the most modest needs of existence governed by rules over which you have no influence – all this is war.

from The Beauty and the Sorrow: An Intimate History of the First World War by Peter Englund, Profile Books, 2011

Ideally, at a local level troops were rotated according to a fairly consistent rota to try to keep men relatively fresh, with a set number of days in the forward trenches, followed by days in the support lines a little further back (mostly spent on work details), followed by rest periods in relative safety further back still, out of the trenches in reserve areas. These schedules would rapidly break down in the case of a major attack by the enemy, or if a big offensive was planned, and men could end up stuck in the front line for weeks.

So in deciding who would be in the first wave of an attack, a number of factors would be in play. At the most senior levels, a preference might be expressed for particular regiments (or even nationalities) to lead the way: some regiments (such as the Royal Welch Fusiliers, with which Siegfried Sassoon and Robert Graves served) had a strong reputation for 'fighting spirit', and Irish regiments had the very dubious honour of being considered good on the attack on account of the Irish status as a 'martial race' - the Australians were also thought of as fine 'shock' troops. Further down the line the Brigade staff would have a better idea of the state of their battalions, and battalion commanders the state of their companies, so decisions could be made as to which unit to put at the front of an attack, which to follow up, and which to hold back in reserve. Beyond that point little control could be exercised, as platoons would be attacking as a whole - there wasn't scope to draw lots or send some men instead of others. Men would be in waves according to their role in the platoon, as this contemporary diagram of a trench-to-trench attack shows; the full text of the infantry training manual that comes from is available here if you'd like to see more. What I've described is from the British perspective but I believe it holds fairly true for all combatants; Germany did develop dedicated 'stormtroop' units to spearhead attacks, but another more knowledgeable person may be able to add more about them.

Later in the war the idea of 'experienced' units began to break down due to the sheer rate of casualties, along with the draft system of sending drafts of conscript men to whatever unit happened to be under strength undermining the bonds of shared community and experience:

There are no heroes here. No one cares. A man is forgotten the next moment. We don’t even know each other’s names, or our officers names, and our officers don’t know us even by sight. We don’t know what we are going to do, or what we are making for when we attack. We don’t know where Jerry is, whether he’s three hundred yards or a thousand yards away. We don’t know anything at all, and we just go forward – every man for himself. Everything is unknown and unthanked, unpraised, and there is no pity anywhere – not even a ‘Poor sod,’ as we pass the men who’ve been killed.

Pvt. R.H. Kiernan, 8th Leicestershire Regiment, 1918

from The Quick and the Dead by Richard Van Emden, Bloomsbury Publishing Plc, London, 2011

So overall, I'd say that the broad answer to the question 'How did commanders decide who had to go first?' is that they would choose (at more senior level) whichever unit was available and deemed capable on paper of achieving the objective, with the amount of choice available and autonomy shrinking as you moved down to more junior levels of command.

Listening to Dan Carlin's podcasts

As a somewhat "meta" aside (which I hope is acceptable to the mods), I'd like to point out that, while Mr. Carlin's narrative style and storytelling ability appeal to a lot of folks (myself included), listeners to his show would do well to take what he says with a grain of salt. If you use the search function, you'll find many discussions (some from this sub, some elsewhere) where the factual accuracy of some Hardcore History episodes is demonstrated to be...shall we say, not up to the same standards of academic rigor that /r/AskHistorians holds itself to. Just bear that in mind that, enjoyable as his show may be, Mr. Carlin isn't exaggerating when he asserts that he, himself, is not a historian.

Carry on...

This is an interesting question and I can not fully answer it, but an important aspect is that soldiers were not just assembled in lines and sent over, but went with their companies in formation, usually split up into half-platoons or sections. The exact formation would differ from battle to battle and sometimes from unit to unit.

So a wave was not, as they might seem in popular imagination, lines of men walking forward, but many groups of soldiers attacking together. Each group would have officers and NCO's to urge them forward, and most likely be made up of a mix of old veterans and new recruits (unless it was a new unit with only new recruits of course). How the dynamic within a platoon or half-platoon would be that decided who moved first I can not answer, but I guess it is a wide question that can be applied to any military unit in any war in history?

Another aspect is that the first wave was by no means guranteed to be the least safe, if we look at the waves as they were (and not just the order of individual men within units). Typically the first wave would capture the first enemy trench-line and consolidate it, while the second wave would pass over them (literally jump over the trenches to keep moving) and make up the first wave for attacking the second enemy line. A third wave would attack the third enemy line (and so on, in extreme cases). So not being in the first wave just means you now have to be the first wave when attacking enemies further back, enemies that will be alerted and probably reinforced before you arrive. Also the faster you can get across no man's land the lower is the risk that you get caught in the enemy defensive barrage, or that their machinegunners have time to scramble from their dugouts after hiding form the attacker's barrage. I think soldiers had pretty good incentives to move as fast as possible and be the first into the enemy trenches, not stay behind.

At Somme 1916 a large part of the British losses actually did not happen in the first wave of units attacking, but in reserve battalions that formed up to advance after them. Once the first wave was attacking the British artillery could no longer keep the defenders under effective fire due to avoiding friendly losses, so when the reserves formed up, sometimes in plain sight from German lines, they came under heavy fire and suffered heavily before even reaching no man's land.

For a source on the Somme and analysis of British losses, see Prior & Wilson The Somme. They estimate that 30% of British losses were in units behind the British front line before they even reached no man's land.

For wave formations there is a chapter about this in the contemporary manual Notes for Infantry Officers on Trench Warfare. It does not go into any details about the formations used though, but I can try to dig up some better source.

Further to the answers already provided it's interesting to note the German army and their approach to trench warfare during the middle and latter part of the First World War.

To get the most out of their soldiers the German military organised their infantry into three different tiers. Some were elderly, unmotivated soldiers who were appropriate for only defensive action. The next were best for defensive with limited offensive action and the troops most used for offensive action (increasingly in the late war period, counter offensives) were the sturmtruppen.

'Stormtroops' were young, unmarried and motivated. They were also well equipped with grenades and light machine guns and most importantly, allowed the most flexibility of any German soldiers down at the section and platoon level.

Source: Stormtroop Tactics: Innovation in the German Army, 1914-1918 by Bruce I. Gudmundsson.