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Utilitarianism is an ethical school of thought with its roots in consequentialism  championed by some of the greatest philosophers of moral ethics alive and to have lived. Utilitarianism although championed by such great thinkers as Jeremy Bentham, John Stuart Mill and more recently Peter Singer and Richard Mervyn Hare does have its critics in some equally influential historic and modern thinkers such as Karl Marx, John Rawls, Thomas Carlyle and more recently Bernard Williams. As with every school of moral ethics Utilitarianism answers questions on moral rights, wrongs and codes of action however some critics would say it also throws up as many questions as it answers. A Utilitarian theorist would say the course of action which should be followed in any given situation is the one which will produce the greatest net benefit. This theory in itself has immediate problems and in consequence Utilitarian theorists have adapted the original idea put forward by Jeremy Bentham giving genesis to many different forms of Utilitarianism such as Act, Rule and Preference Utilitarianism.
As mentioned Bentham was the original protagonist of Utilitarianism as a school of thought and brought the theory forward in an Introduction to the Principles of Morals and Legislation (1789) (from now on An Introduction) however the theory can be traced as far back as Ancient Greece and Epicurus who was a Greek philosopher 341 BCE - 270 BCE. Bentham's principle of Utility in An Introduction is described as the "principle which approves or disapproves of every action whatsoever, according to the tendency which it appears to have to augment or diminish the happiness of the party whose interest is in question. . ."  . In other words Bentham says that every action ought to be judged right or wrong depending how much it promotes or damages the happiness of those the action affects. Bentham believed the rightness or wrongness of an action is determined by the amount of pleasure or pain the action produces. The principle of utility is also known as the greatest happiness principle which promotes the action that should be taken, or refrained from being taken, is the one which would create "the greatest happiness for the greatest number of people"  (first promoted by Francis Hutcheson in 1725)  . Although this exact phrase is not present in An Introduction it is hinted at "Ethics at large may be defined, the art of directing men's actions to the production of the greatest possible quantity of happiness, on the part of those whose interest is in view"  and is also mentioned in other works such as An Introductory View of the Rationale of Evidence (1811)  . Bentham in his opening words of An Introduction states "Nature has placed mankind under the governance of two sovereign masters, pain and pleasure .It is for them to point out what we ought to do, as well as to determine what we shall do. On the one hand the standard of right and wrong, on the other the chain of causes and effects, are fastened to their throne. They govern us in all we do, in all we say, in all we think..."  . Bentham is saying in its simplest form man is governed by pleasure and pain and these two ends are the factors we consider when making the moral decisions in life to maximise pleasure and minimise pain. Bentham also put forward a theory for measuring the moral value of an action in chapter 4 of An Introduction he called this the Hedonic or Felicific Calculus. The Felicific Calculus is used to work out the sum total of pleasure or pain caused by any given action and therefore the total value of its consequences. Bentham said when calculating what action is morally right in any given situation we should give consideration to the pleasure or pain which will occur in respect of their Intensity (strength of pleasure/pain), Duration (how long will the pleasure/pain last), Certainty (how likely is it that the pleasure/pain will occur), Propinquity (how soon will the pleasure/pain occur), Fecundity (chance the pleasure/pain will be followed by other feelings of the same kind), Purity (probability it will not be followed by feelings of the opposite kind) and Extent (amount of people who will be affected)  . Next the alternative courses of action should be considered this will illuminate which course of action will maximise pleasure and create the "greatest happiness for the greatest number"  .
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Bentham and his views on Utilitarianism have critics from those of opposing ethical views as well as other Utilitarian's, Karl Marx criticises Bentham for having a narrow view of the normal man and the fact that the world is ever changing "With the driest naiveté he takes the modern shopkeeper, especially the English shopkeeper, as the normal manâ€¦. this yard-measure, then, he applies to past, present and future"  . Thomas Carlyle famously labelled Bentham Utilitarianism as "pig philosophy"  and perhaps the most surprising of all critics is John Stuart Mill who criticised Bentham on his narrow view of pleasure. Bentham said "the game of push-pin is of equal value with the arts and sciences of music and poetry"  Mill disagreed with Bentham's quantitive hedonistic view of happiness and how he had over simplified human life saying "It is better to be a human dissatisfied than a pig satisfied; better to be Socrates dissatisfied than a fool satisfied"  . Mill is saying Bentham is ignoring the difference between higher pleasures and lower pleasures and his felicific calculation only takes into consideration quantity of pleasure and not quality.
Mill as mentioned believed that pleasure cannot be judged by the intensity and duration of the pleasure alone "it would be absurd that while, in estimating all other things, quality is considered as well as quantity, the estimation of pleasures should be
supposed to depend on quantity alone"  . Mill agreed with Bentham's definition of Utilitarianism "The creed which accepts as the foundation of morals, Utility, or the Greatest Happiness Principle, holds that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness  ". However Mill put forward a more in depth system for judging pleasure which at first glance could be construed as Elitist. Mill rejects the conception put forward by Bentham that pleasures are "interesting perceptions" and sees pleasures as the enjoyment of activities, these activities are to be judged on their value by what Mills describes as the Competent Judge and the activity that the Competent Judge prefers will be the more valuable "From this verdict of the only competent judges, I apprehend there can be no appealâ€¦â€¦the judgment of those who are qualified by knowledge of both, or, if they differ, that of the majority among them, must be admitted as final"  . As mentioned Mill uses a Competent Judge to evaluate pleasure determining this by a system of preference, however a normal person in the street cannot be a Competent Judge, a Competent Judge must be positioned well to judge these different human activities. Mill gives three conditions which must be met to qualify as a Competent Judge. Firstly acquaintance, a person must be "competently acquainted"  or well informed about the different forms of pleasure. The person must be "equally susceptible to both classes of pleasure"  and have the capability of enjoyment of different classes of pleasure. Lastly the person must hold a position whereby he has the ability to make effective choices from a range of pleasures, in essence the opportunity to make the choice. Mill said a pleasure is a qualitive better and not just a greater quantity if there is a "decided preference" shown by all or a majority of the Competent Judges "Of two pleasures, if there be one to which all or almost all who have experience of both give a decided preference, irrespective of any feeling of moral obligation to prefer it, that is the more desirable preference"  . Mill believes the activity which the Competent Judge prefers, the "decided pleasure", would not be swapped for any quantity of the less preferred lower pleasure "would not resign it for any quantity of the other pleasure which their nature is capable of  " which therefore makes it the higher pleasure. Mill goes on to describe what is meant by a higher pleasure, saying "it is an unquestionable fact that those who are equally acquainted with, and equally capable of appreciating and enjoying, both, do give a most marked preference to the manner of existence which employs their higher faculties"  . By "higher faculties" Mill means our sense of perception, judgement and reasoning and he believes that a "manner of existence" where we use our "higher faculties" is preferred by the Competent Judge who meet the three criteria and therefore the mode of existence with those pleasures is superior. This is where Mill could be challenged for being elitist as he bases his judgement of the pleasure with the most value, and therefore the pleasure which is to provide "the greatest good", on the preference of class of a Competent Judge and not what is good for humans generally. Mill didn't believe the Competent Judge to be a higher being and he knew the ordinary man would not have the same experience and would therefore hold a different view on what was more pleasurable. However he believed any person who met the standard of a Competent Judge  would share the same view and come to the same "decided preference", that human nature would take over and a common preference would emerge for this mode of existence. Mill's reasoning an ordinary human would give a different decided preference than the person of higher faculties is threefold and these three reasons correspond with the criteria for being a Competent Judge. Mill believes that sometimes a person will not acquaint himself with the full array of alternatives to that pleasure and many who are capable of achieving the higher pleasure choose to aim for the lower as it is more easily attainable "It may be objected, that many who are capable of the higher pleasures, occasionally, under the influence of temptation, postpone them to the lowerâ€¦.Men often, from infirmity of character, make their election for the nearer good"  . Mill also believes some may lose their susceptibility and opportunity to the higher pleasure due to neglect or a loss of capacity to enjoy possibly due to lack of financial or material resources or health, education or liberty "Men lose their high aspirations as they lose their intellectual tastes, because they have not time or opportunity for indulging them; and they addict themselves to inferior pleasures, not because they deliberately prefer them, but because they are either the only ones to which they have access, or the only ones which they are any longer capable of enjoying"  . In essence Mill says in order to determine what is good and therefore to attain the "greatest happiness" we must choose a Competent Judge who has been in a position to make informed judgements based on knowledge and experience and that any person if given the opportunity would come to the same common preference. Mill also believes that any person with a moderate amount of intellect can, with sufficient acquaintance, susceptibility and opportunity can become a Competent Judge and enjoy the higher pleasures "In a world in which there is so much to interest, so much to enjoy â€¦.every one who has this moderate amount of moral and intellectual requisites is capable of an existence which may be called enviable." 
As mentioned above there are different types of Utilitarianism and all have merits and drawbacks Bentham Utilitarianism can be classified as Act Utilitarianism which means when faced with a choice all the possible outcomes of potential acts must be analysed and from there decide which act will produce the most happiness or least pain for the greatest number affected by it. For this system to work there must be a universal way of measuring pleasure and pain and as mentioned Bentham brought forward the Felicific Calculus which needs no more explaining. The Felicific Calculus can be a long pain staking process which has to be applied to every situation for every person the action will affect, therefore if there are many choices of many situations the system becomes impractical. Bentham himself admits that to use this method for every moral decision would be nonsensical "It is not to be expected that this process should be strictly pursued previously to every moral judgement, or to every legislative or judicial operation." 
Rule Utilitarianism is a departure from Bentham's Act Utilitarianism and works on the principle of choosing the rule which will maximise happiness, therefore instead of analysing the outcome of an act, the rule is followed which will create the overall best outcome. This is done by determining the amount of good, or finding the moral value of following that rule, whichever rule has the best aggregate consequence is the best rule. Rule Utilitarianism has been criticised as automatically collapsing into Act Utilitarianism  , this happens when the rule which has been chosen to maximise happiness, if followed will cause less happiness than an alternative. This will create a choice or sub rule which will be judged on the best overall consequence therefore becoming Act Utilitarianism. Mill said it is sometimes right to violate ethical rules "..justice is a name for certain moral requirements, which, regarded collectively, stand higher in the scale of social utility, and are therefore of more paramount obligation, than any others; though particular cases may occur in which some other social duty is so important, as to overrule any one of the general maxims of justice. Thus, to save a lifeâ€¦"  Mill says even the highest contributor to the greatest happiness can sometimes be reverted from if it is in the interest of a greater good.