Schrödinger's cat is the idea proposed by Edwin Schrödinger. This idea deals with the experiment of placing a cat in a box with a geiger counter attached to a hammer and a vial of poison. For those who don't know, a geiger counter measures radioactivity. This forms the paradox of Schrödinger's cat because the half-life of the radioactive material is unknown. Then, whenever the geiger counter detects a change in the radioactivity, it activates the hammer which smashes the vial of poison killing the cat. The only thing is that we do not know if the cat is dead or alive. This is why we say that the cat is dead and alive because, while it is in one of those states, we don't know until we open the box.
This theory of a cat being both dead and alive connects to the idea of superposition. Superposition is the idea that an atom can occupy multiple states at the same time before it is measured. Because we do not know whether the cat is dead or alive until we open the box because of said superposition principle, this allows the theory to work. However, it is important to understand that this does not scientifcally prove anything. This was simply an example to explain how parts of the quantum theory works