So, yeah.
It won't boot if I don't plug it.
Fortunately, I can still unplug it in middle-work.
I know I can deal with it, but it will annoy me if I keep going that way...
Help would be much appreciated!

Note: This happened since I installed *IOBit Driver booster*, which means yesterday.
Idk if it's a driver update that done this ...

uh, telling us what kind of computer you have would help

I don't really think it has a problem with the computer's type ... but well :
HP 6530b

Well for future reference, yeah, we need to know what kind of computer you're asking us to help you with.

How old is the battery? If you plug the laptop in without the battery, does it still boot up?

YOU SHOULD GET A NEW BATTERY.

Quote from jbutouden

Note: This happened since I installed *IOBit Driver booster*, which means yesterday.
Idk if it's a driver update that done this ...

then try to uninstall that,

Quote from Exilement

How old is the battery? If you plug the laptop in without the battery, does it still boot up?

Must be 1/2 years old.
It does boot up with no battery too.

*New problem*: It does shut down if I shut the screen (laptop) for what seems to be some seconds .

So your saying even if the battery is charged and it's unplugged you cannot boot the computer unless you plug it anyway.
I'm not an expert but i had a similar problem once and it had to do with the switch that's activated when you fold the laptop.