These are not branches as much as they are paradigms or theoretical approaches which historians utilize to devise and answer specific questions about the past.

The way professional and/or academic history operates is that when writing a lager work they usually revolve around a larger question that is derived from the state of scholarship and the author's interest. Because historians are generally aware what the state of scholarship is and what questions remains unanswered or could be taken up anew, they go about answering these questions with a certain research design (what are my hypotheses/theses, which sources will use) in mind.

Here's where approaches like cultural or social history come in. Let's say that the question of our hypothetical paper revolves around the French Revolution, f.ex. what factors contributed to the French Revolution's success. Approaching this from a social history perspective will lead to the question as well as the potential answer being viewed through a different lens than a culture history approach.

A social historian will approach this from a standpoint (broadly speaking) that the factors contributing to the Revolution and its success will be found in social conflict such as economic changes, poverty, class struggle, and famine. Accordingly, what they look at are sources that reflect this approach, such as statistics about the rising bread prices, the development of wages before and during the revolution, hunger in certain regions, and the rise of the bourgeoisie in an economic sense before and during the revolution. They might find that wages stagnated while bread prices were on the rise, that in terms of economic importance the bourgeoisie began to replace the nobility in terms of economic output and that was accompanied by clamoring for greater participation in the matters of state, and that within the manufacturing and production process, modes of capitalist production came to more and more replace other modes of production such as guilds. What social historians tend to focus on are the changes in a social structure – economic, political and so forth – and how they influenced historical development. In the words of one of its most famous proponents, Hans Ulrich Wehler, social history "It stressed structures and processes over actors and events. It emphasized analytical approaches close to the social sciences rather than by the traditional methods of historical hermeneutics."

A cultural historian on the other hand will (while certainly pointing to the above) be more focused on changes in culture around the time of the revolution and their work will resemble that of an anthropologist more than that of an economist. They will look at different sources such as how nobility was perceived in literature, how established cultural practices changed and how that reflects a new understanding of the world, what symbols and practices of the revolution can tell us about the reason for participation and so forth. For example, a classic text in cultural history is Robert Darnton's The Great Cat Massacre of Paris. In it, Darnton recounts the 1730 episode of the printers' apprentices of Paris slaughtering their masters' cats after they held a mock trial against them for being treated better than they were. Darnton's point of this episode is to show that cultural practices of protest against maltreatment took up forms of then established means of social interaction (the cats were tried in effigy like it was popular in 18th century France carnivals) and turned them into means of protest against perceived maltreatment. It is an example text of how cultural historians will take not the economic changes per se – how printers' apprentices were maltreated because of economic changes at that time – but instead focus on the everyday practices of people, their use of symbols and said practices and their individual behavior and how all this changed over time to come to a deeper understanding of what social and economic changes meant to the people living in past societies.

An, say, intellectual historian or a historian of ideas on the other hand will focus more on the – unsurprisingly – intellectual history and the ideas behind the revolution. What they will focus on is how enlightenment ideas from Locke to Hobbes to Hume were read and understood by the historical actors in important political positions and how they tried to implement those ideas, f.ex. how a variety of Enlightenment ideas and scientific advancements contributed to Robbespierre's formation of the cult of reason like it did in the aspect of the metric system being taken as a basis for a virtual new ordering of time.

So, what these approaches present are different foci in how the past is studied and approached. People who employ them will very often, and often also convincingly, argue why the approach they employed hold greater explanatory potential, greater potential for approaching a deeper understanding of the past, than others. In principle however, most of these different paradigms of interpretation all have value in uncovering more about the past. It is on the other hand, correct though to state that they also have limitations when it comes to certain questions and are better at bringing into focus certain aspects over others.

What – almost – all of them share however and why they are so important as a function in how we research the past is that they deliver a framework, a lense through which we can view and interpret history. No one person is able to write the ultimative history of anything. Because we are bound to the medium of text (or language in general), there is always the limitation that a text need to start somewhere and end somewhere. That we need to bring the complexities of a human society and all that entails into a coherent narrative that connects these beginnings and ends in fashion that makes sense. We work to establish cause and effect of events, deeds, and ideas, to draw attention to connections, and extract meaning from past events by putting them into a larger context. This process requires that certain factors be highlighted while others are relegated to the sidelines or not taken into consideration at all. We can't conceivably start a history of the French Revolution by writing about human history from its very beginning. We can't even write a history of the French Revolution by incorporating everything that might be closer to it in time for that would exceed the ability of one person in terms of skills, writing, and researching by a wide, wide margin.

Because of this, we need to rely on a framework to help us determine our focus, to guide us in what questions we should ask, and of how to interrelate and interpret our sources. A cultural historian will see a different relationship between sources and interpret one source differently than a social historian will and so forth.

So, in short: These different "branches", these approaches give us frameworks and methodology to narrow our questions to a sensible level, to help us approach these questions and give us tools to interpret sources. They serve as approaches that will help determine on what we focus and how to approach a deeper understanding of the past.

As you can grasp from trying to follow the conversation online, the subdisciplines of history are more like umbrella terms that have a web of meanings and connotations rather than representing a single set idea. And it sometimes feels like there's been almost as much written about what they are, as there has actually applying them.

Nevertheless, here are the shorthand ideas I generally work with:

Political history - "politics, power, and policy"; polities and their institutions of government and external affairs (laws, leaders, wars, etc).

Social history - identity politics applied to history - thinking about people as Groups and how the Groups relate to each other

Economic history - jobs, taxes, and trade (external and internal; goods and currency), and the people who participated

Intellectual history - dead white men claiming themselves to be scholars, theologians, and philosophers, the heavy books they wrote, and the Big Ideas in those books. Sometimes borrowed by scholars of non-white men as a way of garnering respect for the intellectual work of their subjects, but this scholarship remains treated as peripheral.

Cultural history - shared ideas, attitudes, beliefs, practices, rituals, media - trying to recreate the mental worldview of a person in the particular era in the past, including their perceptions of the above. Basically: the interesting bits.

Today, it's hard to find "pure" histories of one sort or another. Economic history and intellectual history might still have their holdouts. (Generally when they are written by economists and philosophers respectively.) But as you saw already with my note about "political culture," cultural history and social history have brought their influences of cultural anthropology and sociology to the other subdisciplines.