I'm especially interested on how it was organized before, during and after the Civil War (1642-1651). In history class we are studying the 17th century. We saw how the Swedish army begun to organize itself more similarly to modern standards, with trained soldiers instead of normal people or mercenaries. So I would like to know how the English army changed its organization with the advent of capitalism and the Civil War.
That is a very big question and covering a very large period of time. So, my answer will be with four major caveats. Firstly, I'll focus on the 17th century and mostly that of the first two Stuart kings of England - James I and VI and Charles I. Secondly, when I discuss the English civil wars, I will actually be talking about the British Civil Wars, the preceding and ongoing conflicts between and within the Stuart realms of England, Scotland, and Ireland cannot functionally be separated in with cause or effect. Don't worry you'll get your answer, but this covers me if I stumble into talking about the rest of the British Isles (at the time, Wales was still in the 1536 act of Union so considered for political and administrative purposes, but not culturally, part of England). Thirdly, I'm doing this off the top of my head as my books are still in storage, so apologies if I don't quote too much. Fourth and final, your question is a BIG one, so I'm hoping my whistle-stop answer will help frame in generally and you can then follow up on the particulars you want to dig down on afterwards.
Right. England did not have a standing army as we understand it. In fact throughout the period you are interested, pre 1640, the defence of the realm was seen as lying heavily on the shoulders of the militia. Scotland too had some fencible men, but apart from scattered companies of artillerymen in important castles or forts, the bulk of the King's armies were non existent, with three exceptions.
There was a small army in Ireland, but this was mostly at company level, and scattered across in disbursed garrisons, it was also, nominally under the command of the Lord Lieutenant of Ireland, and paid for by the Irish Parliament, while some recruits came from England, it was more an Irish force than English
Both James and Charles had no problem allowing either foreign kings or adventurers raising English troops for foreign service. In fact the ability to provide troops to foreign wars served the crown's interests - rambunctious adventurers, religiously zealous men of fighting age etc were actively encouraged to sod off elsewhere. Commissions would be offered to men interested in commanding such a force, and then he would set about recruiting his regiment. Usually about 1200 men, and then take them away. Often these were so officially sanctioned that the crown would allow limited local in conscription to aid recruitment. This emptied the jail's of felons and conscription mopped up the poor and undesirable folk, usually the unemployed, vagrants, and petty criminals and got rid of them. These foreign forces provided a school for soldiers from England, particularly officers who the crown could call on at need, but not have to pay their wages or employment in the interim.
On the rare occasions the English crown itself went to war, and in these instances regiments were usually raised in just the same manner as 2, but with the commission's going to men selected as colonels by the privy council. At war's end, the regiment would be "paid off" (note, probably not actually fully paid) and the units disbanded.
This obviously had the flaws of not really creating an institution of military experience, but for the cash-strapped Stuart kings had the best of what they needed. And all was well with a land enjoying several decades of internal peace while Europe tore itself apart with religious warfare.
That is, until the crown decided to raise forces to put down the rebellion in Scotland in 1638. The First and Second Bishops wars were a wonderful example of how difficult the English system was the raise a force for the Royal army. Although, contrary to popular belief, the efforts to raise the English forces for the First War bore quite some success, however peace was signed before any real fighting occured and when the king needed to raise a force in 1639/40, because he was basically doing it again from scratch it met with a lot of resistance. Men had already paid taxes the year before, and the most likely conscripts had already been rounded up, marched off, then disbanded. The extensive attempts to recruit at army of 10,000 infantry in the regiments sent north from reluctant militiamen, unhealthy and unwilling conscripts, and with not institutional logistics base met with...poor results.
Once the disaster of the Scots victory at Newburn, the occupation with the north of England, the execution of Strafford, the subsequent Irish Rebellion, the attempts to raise counterinsurgency forces in Ireland, England, and Scotland. Well. That's a whole other epic in itself.
So. The key thing here is the regiment. A man was given a commission as a Colonel, and it was his duty to officer and man his regiment with what funds he was issued. Ideally his recruits were to be equipped from Ordnance Office stores or purchase (again another whole topic).
And that is... really how the regiments on all sides of the civil war were organised - on a regimental basis which were then gathered together under a commanding general. After between 1641 and 1644 there was no conscription in England, relying on volunteers but both sides began heavily conscripting in 1644. The recruits and the methods if their recruitment again, big enough for a whole other topic. But even the "professional" New Model Army was filled by conscription. These men were not given limited enlistments, conscription of the period lasting until the powers that be felt they didn't need you - either war's end came and to save money you were discharged. Or your body wore out from hard service or wounds and you were discharged. Or you died, from any one of a numerous reasons so varied and grim it seems a bit unreal to our modern minds. The rate of deaths in service were quite high - so high that enlistment in an early modern army in any nation was seen as a death sentence by many.
After the Conquest of England, Scotland, and Ireland by the English Parliament's forces, it was seen that none of the three really enjoyed being conquered, so the English Parliament felt it had to keep a significant standing army, based from the New Model regiments, as a garrison. It was from these regiments, that the ancestors of the oldest of today's British army regiments trace their lineage.
I'm sorry for the generalisations, but forgive me for saying your question was rather broad - hopefully if you find anything in my answer that you want to dig down on, I can be more specific but it's a huge topic and one I could ramble on about for quite a while. But yes, very happy to answer any follow up questions and I should be able to do so in greater detail.