I've heard of the weregild in England prior to the Norman conquest, but how exactly did it work? When did it become the standard method of justice/solving disputes?
Weregild appears in the earliest codexes of English law - the immediate post-conversion laws of Æthelberht of Kent from c.597 - which implies that at this point, the custom is already a well established part of customary tribal law. The lexigraphical expansion which accompanies the conversion to Christianity merely allows these customary laws to be properly codified and to then be increasingly expanded and stratified.
The initial laws of Æthelberht set weregilds of 40-80 shillings, but by the reign of Alfred, the legal codex implies individuals with weregilds in excess of 1,200 shillings to be a relatively regular occurence. The later legal codes don't tend to state individual weregilds like earlier codexes, which suggests that these may have been listed in a separate document, perhaps one that could be regularly updated without necessitating a completely new codex. There are, however, clues from which we can infer some values; the weregild of a land-owning Welshman in Alfred's codex is set at 200 shillings, from which we can infer that that of an English freeman would be significantly higher. It's worth noting that the penalty for starting a fight in somebody else's home is 120 shillings. Alfred's codex also notes some of the additional fines available: for a weregild of 200 shillings (perhaps a 'cottager' or 'smallholder' tenant farmer) 30 shillings must be paid to the victim's lord. For those with a weregild of 1,200 shillings, 120 must be paid to the lord.
The Anglo-Saxon system is largely based on restorative justice. The maintenance of law and order is seen as the collective responsibility of the community as a whole, often directed through the person of the ealdorman, thegn or 'hundredman' as necessary, of through the king's officers and reeves. By the tenth century, systems of mutual guarantee such as the Frankenpledge were also starting to appear, and it is likely that, at least within settlements, fyrd garrisons or household troops were also used in law enforcement, or at least in the pursuit of fugitives.
The system of weregild typifies the 'restorative' approach, and at heart is designed for the prevention of feud by compensating a victim or their family. Fines are therefore paid directly to the victim or their family, and only to the king if royal tenants, officials or property are involved, or no family or dependants can be found. The system is essentially designed to provide some resource for a family who may have been deprived of their main provider, or as recompense for stolen property. By the ninth century, the law covers a wide variety of injuries which, while not life threatening, may prevent somebody being able to work. The weregild thus compensated them for that lost income, as well as, as previously seen, compensating their lord for their lost income from taxation or service. Similar compensations were extended for thefts, and even a wide variety of sexual assaults ranging from groping to rape.
The exact punishments for a failure to pay weregild are not always specified. In some cases, non-payment is punished by slavery or the confiscation of property. In most cases though, codexes simply state that matters are left to 'The Avengers'. This doesn't mean that Captain America and Iron Man are going to come swooping in to bring justice, but rather that the avenging family of the injured party, or jndeed the injured party themselves, if they had not been killed, were now allowed to exact whatever revenge they now desired upon the perpetrator. Obviously it was this kind of action which led to the sort of feuds which the weregild is explicitly meant to prevent wherever possible.
Well let us start with some extreme basics. Weregeld is the payment of a fine for restitution in cases of criminal conduct or behavior. Theft, assault, rape, abduction, and so on are all included in law codes that have survived. The payment depended heavily upon the social class of the both the grieved party and the offender. In the law codes that are available the most common method of payment described is in shillings. The level of standardization of weregeld and the specific amounts to be paid is something that will likely forever elude our understanding, the early English kingdoms and their laws are somewhat shrouded and incomplete. The most famous of these law codes from early Anglo-Saxon England come from the Kingdom of Kent and date to the 600's.
The Laws of Æthelberht of Kent spell out the payment required for restitution to a variety of crimes. For example, the first law put forth in the text details that property stolen from the church is to be returned twelvefold. This continues throughout the law code. Unfortunately the amount to be paid is not always clear, in some cases it is quite clearly laid out, such as 50 shilling for murder on the king's land, and in other cases it is less clear, such as when recompense is to be paid twofold following offending the king at a feast, there is no system in the laws as preserved for recording how much particular insults would be valued at.
However these levels of payment also vary by the social class of the offended party. Stealing from a bishop is a lot more expensive to repay than theft from a deacon. This continues down the line as well, rape/seduction of a slave is judged as requiring more payment if she belongs to a certain class or owner than in other cases. So these payments were offered to the offended party in many cases, or to the king himself in certain circumstances. If the offender flees in order to avoid payment, then his family takes up the slack and is still liable for a certain amount of the payment.
The Law of Æthelberht is not the only recorded instance of weregeld payments of course though. Kent and other Anglo-Saxon kingdoms all produced their own law codes on the amount and object of the payments. It remained a fixture of Anglo-Saxon law codes up until the Norman Conquest. However payment of money, or in kind, was not the only method of restitution for crimes. Banishment, beatings, execution, and enslavement still retained a place as a punishment for certain crimes, especially one imagines if the offending party was unable to pay the entirety of the fine or weregeld or if the offender was a slave.
Now what are the roots of weregeld as a system of adjudicating disputes? It stems from Germanic societies that existed on the periphery of the Roman world and during the collapse of Imperial authority in the west these societies ended up in control of large amounts of Roman land. In many cases there would be two separate law codes, one for the Germanic people and another for the Romans still living there, this is seen especially well in Frankish Gaul (I say to denote it from Francia or France). In England specifically, it managed to outlast Roman law in many parts of the country, why this was I think is a little tangential. However suffice it to say, that by the end of the 6th Century it seems clear than in many areas formerly part of the Roman Empire and now ruled by a variety of Germanic groups weregeld was a part of the legal system in place.