Omission means a failure to act
According to oxford law dictionary the word omission means: ‘a failure to act' that means when a person is bound to do or act but he omits to do that or deliberately neglects it, it's called omission.
“A summary of criminal law on omissions: a defendant is only guilty of a crime when failing to act, where he or she is under a duty to act”. J. Herring  Criminal Law, text, cases, and material.
For example if a person passing by a pool saw a child drowning in it and there is no other sources present to rescue him from drowning and that person passes by that pool and leaving the child to die, he commits omission.
There are generally two kinds of duties which a human being has towards his fellow men, firstly he is duty bound by law to act according to law of the time or current statute in force, that is called statutory duties, failing to those he could be guilty of criminal prosecution. On the other hand there are some moral duties of a man towards his fellow human being, in that case he is not duty bound by law but by moral traditions or moral values, these duties are called moral duties.
A legal common law duty is nothing else than the enforcing by law of that which is a moral obligation without legal enforcement. Lord Coleridge CJ in Instan 
In majority of the criminal cases there are two types of element involved, firstly its mens rea which is an intention to commit offence and it's related to the mind of the offender, secondly there is actus rea which also called the act of the offender which is related to doing or acting of a particular act. As far as the omission is concerned it is omitting of an act which a person is bound to do or act, in UK laws there are no specific laws described to punish a person who omit to do or omit to act unless and until he is duty bound to do such act. “This is not the same as owing a duty of care, but generally “where there is a duty to act there is almost inevitably a duty of care”. J. Herring and E. Palser. There are certain people who owe duty towards the others such as mother or father owes a duty towards their children, partners living together is the same dwelling and the duties a doctor owes towards his patients. These are not a kind of statutory duties these are in fact a kind of moral duties because these are not specified in any law or statue.
As the time progresses statues are making new laws on such duties for instance the s.5 of the Domestic Violence Crime and Victims Act imposes a special kind of duties on the people of the same dwelling to safeguard the life of a child or vulnerable adult who lives in them in it. The recent case of R vs. Khan (Uzma) is an example of it. There is lot more to be clarified as far as the duties are concerned
As far as the common law is concerned there is no general duty or criminal liability for a failure to act, but there are exceptional duties incurred upon the defendant failure to which he is liable to criminal prosecution. For instance in section 6(4) of the Road and Traffic Act 1988 a motorist is liable for criminal prosecution if he on demand fails to provide a breath sample asked by a sergeant for a road side check. Interestingly the punishment for such crime (omission) is same as for the commission of it. Similarly where a driver refuses to provide his ware bouts to an officer when his vehicle is involved in an accident or injuries to some person, he is guilty of an offence under (s.170 (4) of Road Traffic Act 1988).
There are five situation highlighted by the statues where a person is duty bound to act otherwise he may be criminally liable for that.
Where A Duty Arising From A Statue
Where a defendant was duty bound by a specific statue or law to act but he willingly does not do so he is liable of an offence. For instance it is an offence under s. 19 of Terrorism Act 2000 where a person willfully conceal or refuses to give information on the noticing a person of some suspicious activities involved in his dealing. The famous case of Dytham  QB 722, D, where a man was taken out of a bar by a bouncer and was kicked to death in front of a police constable, when the people around their asked the constable for helping the victim the constable refused to do so by saying that he was already off duty. The constable was charged with the offence of misconduct for the negligence to perform his duties to maintain the peace. Similarly there is a common example of a motorist who refuses to give the breath samples when asked by an authorized person; he is surely liable of an offence under s. 7(6) of Road Traffic Act 1988.
Where A Person Is Bound By A Contract
A person is bound to act according to law where he has gone under a contract with others. It is his obligatory and statutory duty to act according to law. He is answerable to the party under a statue to act according to law, failing which he could be criminally liable of an offence. for example the case of Pittwood (1902) 19 TLR 37 where he was hired by the railway company to look after the railway crossing and shut the gate when the train approaches the crossing to avoid any damage or harm to the road users. While one day during the duty he went out and the gate remained open while the train approached and a man was killed while crossing the train track, Pettwood was prosecuted for gross negligence manslaughter.
3. Where A Duty Arises From A Relation
There are two types of relation among the people on is by blood and the other is by subsequent relations. Usually parents are duty bound to take care of their children, this duty is both imposed by law and by morality on them, and therefore they are under a duty to protect their children from any kind of physical harm. In the famous case of Gibbons and Proctor (1918) 13 Cr App R 134, where both man and the woman were charged guilty due to starving the child to death by stopping to provide it food, the man was charged because he was under a duty to take care of his child while the wife who took money from the father of the child automatically owes a duty towards the child to look after it.
4. Reliance Or Voluntary Resposibility
If a couple takes a duty of care for a child to look after him and they are being paid for it that is called contractual duty on the other hand where a couple assume a duty of care towards there vulnerable relative they assume a duty of voluntary responsibility to look after him and take care of him, failing to do so they may be charged for criminal offence. This rule was followed in the case of Stone and Dobinson  2 All E.R 341. In this the defendant was liable for a criminal gross negligence for not taking care of their infirm relative. What would have been the result, had the deceased been not their relative? “The cases involved assumption of a duty in respect of a duty, but there is no logical reason why the assumption should not be in respect of a non relative” A. Reed and Ben Fitzpatrick; Criminal Law  p 31.
5. Duty Arising From Commision Of A Fault
For example if D is has created an unlawful situation without any mens rea but still he holds a responsibility for such act, he deliberately refuses to overcome or ask for any help to get the situation out of danger, he is surely liable for that negligence. Same was the rule followed in the case Miller  2 AC 161. Where D was a vagrant who was tenant in a house, one night he accidently forgot to unlit his cigarette before going to deep sleep and after some time he got up and realized the mattress was on fire, he never cared for it and went to sleep in the other room, the house caught fire and over all a heavy loss was done to the house. Miller was charged for criminal liability. It was held that he would have easily called for help once he got up and realized that his negligence is going to bring some damage to the house.
Should Omission Be Punished Or No? If Yes, In What Circumstances?
It's been a long debate between the authors whether the omission should be punished by law or not. Many of them have argued in favor of it and many are going against the idea to punish omission. It's not been an easy task to talk on such a complicated issue of law. Authors who support the idea of penalizing omission suggest that there should be a clear line between omission, commission and act. It is fact that there are some circumstances where one could be liable for punishment for the wrong doings of another. The rule of vicarious liability is a good example of it, where an employer of a company or firm is liable for the wrong doings of the employee of the same company. Same is the case in the cases of abetment where a person who abets the crime is liable for the commission of the crime of the other.
Andrew Ashworth is a strong supporter of the punishment for the omission but at the same time he put some limitations on the punishments of omissions. Ashworth talks of two different views on omission, firstly the conventional view, where he suggests that criminal law should not punish people unless and until in clear and serious cases. Secondly, the social responsibility view, where he is of the view that there are times and situations where people help one another and further says that individuals need others, and people living in the same society does owe some duties to others living with them. Further he describes three more duties on the basis of social responsibility, the duty to assist those in peril, the duty to take reasonable steps towards law enforcement, and the duty to ensure the health and welfare of one's children.
Similarly William Wilson is of the view that ‘what is morally worse/causally more significant: shooting a child to prevent the agony of her burning to death in a flaming inferno one is powerless to prevent, or failing to save a him from a similar fate by the simple mechanism of unlocking the door behind which she is trapped?' W. Wilson  Criminal Law.
All of the current authors who support the theory of punishing omissions are agreed on the point that there should be certain circumstance where a person should ensure his safety first before helping others; he should not put his own life in danger to save others. The moral duties should always be present in the society for the mutual relations among the people. It has been prove by the recent cases and some of the previous that murder and manslaughter can be committed by omission. The famous case of R vs. Evans  CA, where an important issue of duty of care was aroused. In this case the defendant was the elder half sister of the victim whom she supplied the dose of heroin which caused the death of the victim, the mother of the victim was also present there at that time, when both the defendant realized that the victim was unconscious they instead of calling for medical assistance went o their rooms and watched TV. The victim a 16 year old died of over dose. The defendant was charged for four year in prison and her mother was for two. The question was raised whether the defendant owed a duty under those circumstances after she supplied heroin to the victim? It was held ‘yes' that the defendant owed a duty of care in those circumstances.
As far as authors who support omission not to be penalized they claim that it is a moral duty more than a statutory duty. Some of them say that not everybody has a duty towards everybody. For instance Eric Mack, 1980 argues that “when a person walk pass by a child who is drowning in a pool should he be liable for the death of that child? Using ‘but for' causation he can be argued that had the defendant not been there the child would still have died at the same time and in the same place. The omission failed to alter the status quo.” Many others claim that omission is just a matter of common sense and does not need any statue or punishment for it, and omission can not commit a result, and unless and until a duty to act is not there, the commission of omission is out of question.
G. Williams in on the point that “A crime can be committed by omission, but there can be no omission in law in the absence of a duty to act. The reason is obvious. If there is an act, someone acts; but if there is an omission, everyone (in a sense) omits. We omit to do everything in the world that is not done. Only those of us omit in law who are under a duty to act.” G. Williams  Text Book of Criminal Law, p. 148-149.
The centre point is that there should be a statutory duty, for which one is liable for punishment, is very much difficult to punish one for not compiling with the moral duties, because moral duties are all was exceptional. Everybody in the society is free to practice his own wills within a limit.
G. Williams opposing Ashworth's scope of liability for omission in a way that the courts must b very clear in deciding these cases, and it should be done in clear and open language. He talks in favor of not punishing the omissions and saying that, ‘It is extraordinary to propose that the criminal liability of omitters should be made more stringent than the criminal liability of doers; and I have criticized the proposal on this ground.' G. Williams, (1991) 107 LQR86.
It has been proved by the cases and recent verdicts of the courts that omission in some circumstance can lead to some serious crimes, like murder and manslaughter. There are cases where we can say that omission gives birth to gross negligence and it needs to be brought in a circle of statue and there should be some proper laws and statue to tackle with this type of omission. It can only be made possible making clear and bold laws where failing to act should be punished. The parameters of the duties should be much clearer. In the cases relating to close relation there should be a line drawn between who owes duties towards whom? It would be very much difficult to punish all the by standers who watched a child drowning, so the criteria of punishing the people for not doing should be handled carefully. In the complicated cases of omission it should be the court who decides the duties or negligence. In each and every case the punishment for the omission should be dealt with outmost care.