The word General in General Relativity refers to the fact that you deal with the note general case of no global inertial frames.
You might be used to Newtonian mechanics or Special Relativity where can can pick an inertial frame and talking about constant velocity in that frame and acceleration and force in that frame and the frame covers everything.
But there is no law of physics that says a single inertial frame has to be able to work for arbitrarily times or for things super far away. And in fact in general relativity it doesn't.
The general idea is that if you had a small low mass,low energy, non spinning particle that wasn't feeling an electromagnetic forces or contact forces and so on. And then you let it move freely, then at each place it was at and each time you could make a really really small inertial frame around it that worked just for a small region and just for a small time interval. And in that frame the particle would be at rest.
It's similar to how you could walk on a great circle around a spherical earth and locally you could find a system where everything looks pretty flat and it looks like you are moving in a straight line in a flat space.
So if you choose a frame near the earth it would see the earth rushing up and accelerating upwards. And same thing if you were in the atmosphere.
In Newtonian mechanics the atmosphere has a pressure gradient, the pressure below a layer of air is a bit bigger than the pressure above that layer and the net difference would accelerate the air upwards except gravity pulls it down and so it doesn't go anywhere.
In general relativity you have a purely local inertial frame and there is still a pressure difference so the air does feel a larger pressure from the air below than from above and so it just accelerates upwards. Nothing mysterious, the air feels an imbalance of forces so accelerates upwards.
But that test particle is, by definition, not feeling any forces so in particular it isn't feeling that pressure. Which means it doesn't accelerate upwards with the air so the to someone that wrongly through the air was at rest they thing the test particle is accelerating downwards.
Now when you try to make a global frame that covers both sides of the earth (inertial frames can't be that large) then you have to use a non inertial frame and so you make up fictitious inertial forces to explain why things don't follow $F=ma$ type equations. Just like when you use a rotating frame.
So the frame with both ends of the earth isn't an inertial frame and so has fictitious inertial forces and says things are at rest when in a true inertial frame they are accelerating. But the true inertial frames are local. If you assumed the inertial frames must be global then you actually wouldn't get gravity.
We see gravity, so the inertial frames must be local and only cover small regions. When your frames are local they are like buying lots of maps, you can work in a map and before you get to the edge of your map you should switch to a different one. And as long as you have a Mao of your starting point and your end point and you can always switch to a new map before you get to the absolute edge of one of your maps then you have enough and it works fine, you don't actually need a global map, its nice but not needed. And for instance with the surface of the earth it is hard to map a global 2d map, even though locally you can.