Great question! Despite that we can see gravity when we drop an object and it falls to the ground and feel gravity by not floating into space when we jump, gravity is still technically a theory.
Both Newton and Einstein offer different theories explaining gravity, but we will focus more generally around your question.
Anything with mass also has a gravity, and objects with greater mass have a greater gravity. Distance is also involved, as gravity weakens with distance. So Earth’s gravity comes from its mass causing a gravitational pull on all the mass in your body. That gravity pull is equal to your weight . So if you were on a different planet with less mass than Earth, you would weigh less there.
In the context of the universe, gravity is what holds the earth in orbit around the sun. Gravity is also how our universe formed, pulling space dust together to create stars, planets, and other space bodies. The moon’s gravity pulls on Earth’s oceans, causing tides. Earth pulls on the moon, keeping it in orbit around Earth.
Earth's mass is 5.9724 x 1024 kg and the moon's mass is 0.07346 x 1024 kg. This equates to a surface gravity on the moon of 1.62 m/s2, or 0.165 of Earth's 9.8 m/s2 gravity. The atmosphere around Earth does not make its gravity weaker. In fact, Earth's gravity contributes to how there is even an atmosphere. The gases and particles that make up the atmosphere are kept around the surface of Earth due to its gravity preventing the atmosphere from being lost to space.The moon actually has less gravity than the earth, but not because of the lack of atmosphere. It's because the moon has much less mass. Mass is just a measure of how much stuff is there in something, so a full-sized truck has more mass than a human. Earth has a lot of mass, 5.97*1024 kilograms, or 96,000,000,000,000,000,000,000 times the mass of an average adult human! The moon only has about 1/80th of the earth's mass, so it has much less gravity than the earth. Earth's atmosphere is only a tiny part of its total mass, so the atmosphere doesn't contribute a lot to the earth's gravity.
Now what is gravity? It is a force that any one thing with mass can exert on any other thing with mass. A force is simply a push or a pull in any direction. When you throw a ball, for example, you're exerting force on the ball. We don't typically think about this, but the floor in your house is continuously exerting force upwards onto the furniture so the furniture doesn't fall through because of gravity! With gravity, two things with mass will want to move toward each other. However, we humans don't feel our gravity pulling on another person because it's not very big, but we do all feel the pull of Earth's gravity all the time - we're not all floating in the air, because that would be happening without Earth's gravity!
Earth's gravity is the reason we fall onto the ground, the reason we can have activities like skydiving, and the reason that it's so miraculous that we can fly in planes.