Newton's cannon
Look at any video from the International Space Station and it seems obvious: up there, there's no gravity. Pens drift, water forms wobbling spheres, astronauts sleep standing up. But the station orbits only about 400 kilometres above the ground, roughly the distance from Amsterdam to Paris. At that height Earth's gravity is still about 90% as strong as it is where you're sitting.
So why does everything float? Because the station, and everyone in it, is falling. All the time. It just never hits the ground.
Isaac Newton explained this with a thought experiment, published in 1728, a year after his death, in A Treatise of the System of the World. Imagine a mountain so tall that it pokes out of the air, with a cannon on top firing sideways. A slow shot curves down and lands nearby. A faster one lands further away. Fire fast enough and the ground curves away beneath the ball as quickly as the ball falls toward it. It never lands. That is an orbit.
Try it. Our mountain is as tall as the ISS's orbit, and like Newton we ignore the air. Drag the arrow at the cannon (or use the slider) and fire.
Below about 7.6 km/s every shot crashes. Right around 7.67 km/s the ball circles the whole planet and comes back to the cannon from behind, about 92 minutes later. Go a bit faster and the orbit stretches into an ellipse that swings far out and comes back. Somewhere near 10.8 km/s from this height it stops coming back at all.
Here's a neat coincidence of numbers. In one second, anything dropped near the ground falls about 4.9 metres. And Earth is curved just enough that over a horizontal distance of about 8 kilometres, the surface drops away by roughly 5 metres.
So if something travels sideways at about 8 kilometres every second, it falls 5 metres, and the ground has also dropped 5 metres. It's exactly as high as it was a second ago. Repeat forever and you're in orbit. It's falling the whole time; it just keeps missing.
What you feel as "weight" isn't gravity itself. It's the floor, chair or bed pushing back up on you. Stand on a bathroom scale and it measures that push. If the floor fell away at the same rate as you, nothing would push on you and the scale would read zero.
That's life on the ISS. The station, the astronauts, the pens and the water all fall around Earth together, at the same speed, along the same curve. Nothing presses on anything else, so everything drifts. NASA calls this microgravity rather than zero gravity, because tiny effects like the thin traces of air at that height still nudge things slightly.
You can get the same sensation on Earth, briefly. Planes nicknamed "vomit comets" fly steep arcs so that, for about 20 seconds at a time, the plane falls along the same path as its passengers. Everyone floats, until the pilot pulls up. The ISS just does it without end.
This is also why rockets don't simply go straight up. Reaching 400 km of height is the easy part. The hard part is the sideways speed: most of a rocket's fuel goes into reaching those 7.7 km/s, so that when it gets high enough, it can keep falling around the planet instead of back onto it. Watch a launch and you'll see the rocket tilt over soon after lift-off for exactly this reason.
And the thin air at 400 km does matter over time. It slowly drags the station down, so every so often an attached spacecraft fires its engines to raise the orbit again. Stop doing that, and the endless fall would, eventually, end with the ground.