Android - What does rooting, bootloader, twrp and recovery mean?

Bootloader

The low level implementation of a system which can be used to flash software onto your phone's partitions. Often used to flash root privileges or Stock Roms.

Recovery

The recovery is another system used to manage/wipe/reformat your partitions. TWRP refers to a custom Recovery which allows you to install custom roms in a zip format. The recovery follows the directions which are contained in the rom zip file.

Rooting

Rooting describes the process of getting administrative privileges on your android phone. You need those priviliges to mount your system partition as read and write so you can modify the files and settings. Most of the time you will have to boot your phone into the bootloader and then flash a root file (like the one from ChainFire) from your computer but there are also apps which can enable the root access by a simple click for certain devices.

If you have further questions to a specific device then you can head to the xda forum for your device and read. It's the most important thing to do before attempting to mess with your phone's software.

Ahh, since you have no knowledge regarding rooting and all I'll just explain in the simplest form. Imagine that you have bought a laptop from HP which Congress preloaded with a windows installation and some pre-stuff, I assume that a normal person would't be bothered by it and the extra useless stuff they install on your laptop. They prevent you from uninstalling them, blocks access to C directory and pretty much other things that some medium user would need. So what he does is install a new os, any os I mean windows 10 or 8 or linux. And so he removes all the bloats and gets acces to many things.

Now getting to the point, that to install a new os, or rom, in android's case, you first need to unlock it's bootloader. Now the bootloader does what it says and manages your boot preferences, just like the bootloader and bootmanager in laptops. As it doesn't need to be done in a computer or laptop, it is neccesary to unlock it to install a custom recovery for your phone.

Twrp is a custom recovery like cwm, and benefits of having a custom recovery are many, just like your windows recovery menu which let you reset your laptop or factory default it which just refreshes the window as it was just installed. So a recovery in a phone is just like the PC one, but having a custom recovery is necessary, as it provides a lot more features.

Now as got my meaning of bootloader and recovery, and if you still don't get something, you can just ask.

And now comes rooting as unlocking bootloader and installing a custom recovery doesn't need rooting your phone. Rooting your phone allows you to get admin privileges, just like in windows, and now you can even tamper with the internal files. You can root your phone with the android you got installed with your phone and don't need to install a custom recovery and all.

As I previously said that these things needn't be done for a laptop, but a phone needs it to be done in order to install a custom rom, or rather saying an unofficial update, or an official update which isn't available by some reasons. So that's why you need to do these things.

And by the way, could you tell us which phone are you trying to update, as then we can check and tell you if you can update without going through all this hassle.