Robert K. Merton
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Robert K. Merton (1910-2003) was an American sociologist whose primary focus was on functionalism and deviance. Born in Philadelphia to immigrants from Eastern Europe, he went to Temple University, graduating with a degree in sociology in 1931. Afterwards, he moved on to Harvard University to earn his doctorate and worked under Talcott Parsons, working under him and eventually became the first sociologist to earn the first National Medal of Science. He was a professor of sociology at Columbia University, focusing mainly on deviance and teaching many people who would become prominent sociologists in their own right such as Peter M. Blau and James S. Coleman.
Functions of a society
Merton sought to develop the ideas of how a society functions, and what types of functions reside in one. He found that not all functions held the same form in a society, and that instead there were core, underlying functions that implicitly influenced behavior and showed long-term trends, and that there were the obvious, overt functions that held immediate observable effects on society.
A manifest function is one which effect is overt, obvious, and immediately apparent. For instance, a backpack ban in a school has the manifest function of removing backpacks from the system, changing it to something else. Another manifest function could exist via taxation, where one pays more money to supposedly fund something. Manifest functions are usually easy to identify, trends that clearly correlate, and in general are easily recognizable. In addition, a manifest function is also one which is intended to happen due to the structural or systemic change. In essence, it describes easy to diagnose and predict happenings within a given society.
A latent function, on the other hand, can form in multiple different ways. Either a latent function is one that is unintended by the societal change or shift, such as students moving to using suitcases instead of backpacks. These are commonly referred to as 'side effects' in lay language, and can also be recognizable as events that accidentally occur. The other kind of latent function is one that is not intended or recognized by a majority of society. For instance, excessive taxation with the purpose of funding certain things, without oversight for how much one can be taxed in a taxation system, could accidentally take away funds from someone when they need it within their immediate life. However, if that amount of things are negligible it may be ignored on a wide scale. It is both an accident and not recognized or noticed by the large part of society.
It's usually these that people are talking about when they make claims about social justice.
Emile Durkheim's theory of anomie was a useful one type of strain theory, but it was not very detailed. Merton expanded on this idea by postulating a few different types of deviance and anomie that one can experience based on the definition of the social world around oneself. Merton mostly sought to find the link between 'personal troubles' and 'macro-society', rather than hyperfocusing on either face-to-face interactions or overarching social structures; he instead wanted to give more explanations about the decisions one makes in the face of society. Strain theory is the attempted explanation to the different avenues that people in a society take in relation to the system around them, and how they cohere or clash with it.
The first avenue of strain theory is conformity. Essentially, the person conforms their actions towards the acceptable means to achieve the acceptable goal. In a capitalist economy, for instance, the general consensus is that one works and invests their capital to acquire the goal of continuous profit. To accept these values and to act as if they were your own interest is to conform.
The second avenue of strain theory is innovation. While the goal they pursue is the acceptable one, like in a feudalistic system working for the noble that works for you, the means that one takes is not the acceptable one. For instance, in said feudalistic society one is contributing resources to the noble in charge of them but they themselves institute secretly some illicit good trade or even assassination to circumvent costs for their work.
The third avenue of strain theory is ritualism. When a person feels that the goal above them is unattainable, they may decide to manifest their own goal in front of them to cope. For instance, in a socialist society, the goal of maintaining a functioning society through owner-cooperation may be something they are not capable of doing, therefore they may stagnate their position going through the motions of everyday life but no longer attempting to change their lives or contribute to the system beyond working their means. In essence, the means in their lives becomes a goal in it of itself.
The fourth avenue of strain theory is retreatism. This is where the person rejects both the common means and the common goal presented to them, and instead takes a route that is socially despised by the general consensus. For instance, in a hunter-gatherer society, the common goal of crop harvesting and hunting animals to present food to the community would both be rejected, and perhaps the person in question would instead decide to a) pursue a life in a cave, apart from society, retreating from it, or b) appealing to a nomadic experience, deciding to move away from the community and find their own way. In each circumstance, the person in question is rejecting the given means and goal to do their own thing, with no interest in changing the system.
The fifth and last avenue of strain theory is rebellion. Similar to retreatism, the individual in question rejects the general consensus of means and ends. However, unlike them, they develop their own means and ends and make efforts to change the society or system in question to reflect those means and ends. This is the only avenue that actively seeks to change the system around them. One classic example of this is Martin Luther King, who sought to change society by acting in ways that were not socially acceptable, for a goal that was not socially acceptable, namely protesting for equal rights. However, over time and because of the success of the protests, it became an acceptable means and ends and from then on was considered an acceptable system change.
The Matthew Effect
One last concept that Merton conceived of was the Matthew effect, titled by the same name of the bible verse appropriate to it. Essentially, the core idea is that when a scientist makes a discovery, further discoveries they make have a higher likelihood of also being discovered, contributing to that scientists name. Conversely, a non-well known scientist will not get credit for something they made due to their removal from the public spotlight. In particular, this happens when a prevalent scientist gets credit for something they didn't initially invent or observe, but instead assimilated it into their list of discoveries, whilst removing credit from the person that initially invented or observed the significant scientific discovery. In essence, it describes a positive feedback loop where, 'the rich get richer and more credit while the poor get poorer and don't get credit'. 
Merton discusses some role theory as well. As he observes roles being functional in the veins of society, he notes and critiques previously constructivist role understandings by noting that in reality roles tend to be more complex and flexible towards the situation and the context of the role itself. Not only does the content of the role matter, but also the frequency of the role, the position of the role, the density of the role, who the role is interacting with, and so one. The role's definition is not as unidimensional as it seems, and is in fact an ever-changing structure dependent on the situation that is around it.