Schrödinger's cat is a thought experiment designed to show how certain interpretations of quantum mechanics lead to counterintuitive results.
In the experiment, a cat is placed inside a box with a vial of poisonous gas. A mallet is set up such that it breaks the vial of gas if a particular radioactive atom decays, killing the cat. Since the radioactive decay is a quantum system, whether the cat lives or dies is determined by quantum mechanical behavior. This leads to the conclusion that before the box is opened, the cat is simultaneously alive and dead.
Erwin Schrödinger originally proposed the idea as an absurd example showing that the Copenhagen Interpretation of quantum mechanics - the most popular philosophical interpretation at the time - could not possibly be true . However, it has lived on as a thought experiment fueling both physical theories and the popular imagination.
The two main ideas behind quantum theory are the idea of quantized, or discrete, states, and the idea of superposition.
A physical system is defined by the set of possible states in which it can be observed. For instance, an electron can be observed either in a spin up state or in a spin down state, but never a combination of the two. Similarly, a particle emitted in radioactive decay is observed to be either emitted or not emitted, never partway emitted. However, particles prepared in identical ways will not always be observed to have the same state. Identical uranium atoms will decay or not decay at random times, though the more time has passed, the more likely the atom is to decay. The state of the system will change such that it becomes more likely to decay.
To determine how states change over time, the idea of a superposition of states is required. A superposition is a vector addition of two states. Quantum states can be in any superposition of the observable states in the system. For instance, an electron can be in a state that is 50% spin up and 50% spin down. As time passes, the electron may change states, perhaps smoothly oscillating between 100% spin up and 100% spin down, passing through the 50/50 state at each time. The Schrödinger equation, the analog of Newton's laws for quantum mechanics, describes exactly how the electron will transition between these different superpositions of states.
But this seems to be at odds with the previous statement. How can a system be an a superposition of states if it can only ever be observed in one state? The Copenhagen interpretation of quantum mechanics holds that when an observer observes a system, it "collapses" from a superposition of multiple states down to a single state in a probabilistic way. This is distinct from the smooth changes in superposition that happen due to the Schrödinger equation.
When Schrödinger's cat is observed, it is either alive or dead. But when it isn't, it is in a superposition of alive and dead.
Technical note: the space of valid states is the set of complex vectors with an eigenbasis given by the observable states and magnitude 1. For linear combinations of eigenvectors, the probability of observing each component of the state vector is given by the magnitude of the coefficient of that component. Linear algebra is important for a deep understanding of quantum mechanics: most results come directly from the properties of operators on complex vector spaces.
The collapse interpretation has an issue, though. Instead of just you observing a cat in a box, imagine putting yourself and the cat in a room and having your friend wait outside the room. You run the experiment, open the box, record your observations, and only then have your friend open the room and see what your observations were. From your perspective, the cat is in a superposition of alive and dead until you open the box, at which point a collapse occurs. You are then in the definite state of having seen the cat, a state which persists until your friend opens the door. But from your friend's perspective, up until they open the door, you are still in a superposition of having seen the cat dead and having seen it alive.
Even worse, this setup can be repeated again and again, such that every new observer is placed in a larger room. Observer \(n\) always thinks that the state has collapsed before observer \(n + 1\). The idea that different observers will disagree on the state of reality in this experiment is problematic.
The many-worlds interpretation of quantum mechanics solves this problem by rejecting the idea of collapse entirely. It instead claims that there is always a superposition of two "world-branches," one where the cat is dead and one where the cat is alive. When you open the box, there is now a superposition of two worlds with two versions of you. In one world, the cat is alive and you see the cat is alive. In the other world, the cat is dead and you see the cat is dead.
 Schrödinger, Erwin; Translated by Trimmer, John. The Present Situation in Quantum Mechanics. Proceedings of the American Philosophical Society. Retrieved on 7 Mar 2016 from http://www.tuhh.de/rzt/rzt/it/QM/cat.html
[flickr] Flickr user chwalker01. Retrieved on 7 Mar 2016 from https://www.flickr.com/photos/31690139@N02/2965956885