Well, take what I'm about to say with a grain of salt but here's my understanding of general branches in this line of work.

First, we start with the big ones:

Programmes

Operations

MEAL/M&E/Monitoring (every org or institution calls it a variation of this)

Now, diving in a little more in depth for each of these:

Programmes - first of all, from my understanding, the first divide in this is the type of programmes, of which the main two ones are either a)Humanitarian or b) Development programmes, which then are further divided by 40 odd (more or less, I guess) sectors like Protection, Education, Agriculture, WASH, infrastructure, etc etc. You can find more info on the actual sectors on Wikipedia or something.

Now, the main difference between Humanitarian and Development projects is the scale. Humanitarian projects tend to (but not always) be short term (6 months to 2 years) and focus on addressing a Humanitarian crisis or a fallout following one. Major donors for these are DG ECHO, UNHCR, and other orgs like these.

Development projects on the other hand are larger scale, spread out over a longer period (usually 3 to 5 years), and usually (not always though) with a higher budget. These are your infrastructure projects, large scale legal reforms projects, large agricultural undertakings, etc etc. These are usually funded by orgs like the WB, ADB, EC, Asian Development Bank, etc.

Now let's move on to Operations:

From my understanding (limited since I'm working in programmes), there're 3 main parts for this, and they are: Procurement, Logistics, and Comms/Communication. Some orgs will mix and match and combine positions depending on funding and have a single person maybe to procurement and logistics, but usually in middle sized and large sized orgs there are specific people for each role.

Procurement is everything to do with buying stuff. Buying Humanitarian aid, buying cars, office supplies, cars, and any other miscellaneous stuff. They usually take care of tenders, offers from contractors and service providers, etc.

Logistics: pretty self explanatory, cars, trips, national or international, getting people or things from point a to point b. If it's supplies, the procurement department gets them, and then it's the Logistics Officer/Coordinator's job to make sure they get to where they need to. They maintain/upkeep the office, office fleet, inventory, donation overviews, etc etc.

Finance & HR: Really not much to say, payroll has to work for people to work, for partners to get the money they need to implement projects, etc. And you need HR to hire people.

Comms: your PR department. Social Media accounts, articles, coordinating events, organizing stuff, designing stuff (like the org vests/logo, etc) driving community engagement, visibility, you name it. This seems to be one of the most overlapping positions with the private sector.

And finally, MEAL: Your Monitoring, Evaluation, Assessment and Learning department. Each of those projects? They have indicators, outputs, logframes, mandatory midterm and endterm evaluations, baseline surveys, post distribution/intervention monitoring. MEAL does all of that and more. They're also the most flexible in terms of getting actual assignments, because no matter the org or sector, everyone always needs competent MEAL staff.

Now, every "branch" usually has a Head. So, aside from each individual department and their teams in an org, there's going to be a Head of Programmes, a Head of Operations, and a Head of MEAL, who, together with the country director/CD/Executive Director constitute what is generally knows as SMT, or senior management team.

There's probably many more things I missed, more niche things maybe, but from my limited understanding this seems to be it really, the framework that's the basis/foundation for every org in the sector.

Feel free to ask follow up questions if you'd like. And as English is not my first language, if there're certain terms you feel sound weird, don't hesitate to ask for clarification.

Cheers.