The atmosphere is the air that is wrapped all around a planet. Not all planets have atmospheres. In order to have an atmosphere, the planet has to have enough gravity to hold on to light atoms like hydrogen and helium and keep them from floating away into space. That means that the planet has to have a lot of mass.
Because the force of gravity is stronger near the planet and gets weaker as you get further away, the atmosphere is thicker close to the ground and gradually gets thinner as you go further out into space. There is no sharp edge to the atmosphere.
Earth, our own planet, has an atmosphere. When the Earth first formed, about four and a half billion years ago, Earth's atmosphere was almost entirely made of hydrogen and helium atoms, because they were the lightest atoms and floated to the top. But the Earth was still so hot, and the Sun heated the Earth so much, that most of the hydrogen and helium atoms ended up drifting off into space.
A volcano in Costa Rica
Soon after that, about 4.4 billion years ago, the Earth cooled down a lot. But there were still a lot of volcanoes that shot out steam, carbon dioxide, and ammonia. This created a new atmosphere made mostly of carbon dioxide and water, with some nitrogen.
Then about three billion years ago, early prokaryotic cells, one of the earliest forms of life on Earth, began to use photosynthesis to get food for themselves. They made their food out of what was available - sunlight, carbon dioxide, and water. And they excreted (pooped out) what they didn't need - mainly oxygen. At first most of these oxygen atoms bonded with other atoms to form molecules, like this iron that has turned red by combining with oxygen.
Iron that has turned red because of oxygen in the air
After almost a billion years of millions of cells shooting out oxygen, everything that oxygen could join with had enough oxygen, and the leftovers began to pile up in the atmosphere. Quickly there got to be a lot of oxygen in the atmosphere, or in the air. By 2.2 billion years ago, the atmosphere was about 20 percent oxygen. We can see this early oxygen in old rocks, where about three billion years ago the iron in the rocks begins to be red from combining with the oxygen in the air (rusting). All this oxygen poisoned many kinds of early cells and they died off. But other cells evolved to use oxygen for their energy, and these cells also benefited from the development of an ozone layer, a layer of one kind of oxygen high in the atmosphere that keeps most of the Sun's ultraviolet light from reaching the Earth and causing sunburns.
Since that time, the levels of these atoms in the air have changed from time to time, even though the atmosphere has continued to have a lot of oxygen in it. About 500 million years ago, the atmosphere was about 7 percent carbon dioxide. That was good for plants. About 300 million years ago, carbon dioxide was about where it is now, less than 1 percent. At about the same time, oxygen went up to about 35% of the atmosphere. Then carbon dioxide went up again, so that about 100 million years ago, the atmosphere was about 3% carbon dioxide. Big forests of giant ferns grew up because of all the carbon dioxide in the air, and the Earth got so warm that dinosaurs could live near the South Pole. Oxygen levels have also gone up and down over the last two billion years, though we don't understand the changes as well.
Today, Earth's atmosphere, or air, is about 78 percent nitrogen (mostly from the ammonia shot out by volcanoes), 21 percent oxygen (from photosynthesizing cells, mainly one-celled algae in the ocean), and less than one percent each of argon, carbon dioxide, and water. But because people are burning so much oil and coal that are made of carbon, we are releasing a lot more carbon dioxide into the air, and the percentage of carbon dioxide is going up. Right now the percentage of carbon dioxide is higher than it has been any time in the last 650,000 years (since the earliest people were beginning to leave Africa and travel around the Earth), and it is still going up quickly. This will cause the Earth to get warmer. Nobody knows exactly what this will mean for plants and animals on Earth, or for us.