How can Schrödinger's cat be both dead and alive?
Before reading this answer (and to those who are downvoting), I am addressing if the cat is both alive and dead. I don't think the question is asking for a complete explanation of the Schrodinger's cat experiment, nor is it asking how this links to all of the deeper mysteries of quantum mechanics and how we should think of them. Therefore, while there is much to be gained in thinking of many different interpretations, I will not be addressing them here.
Schrodinger's cat is not both dead and alive any more than an electron simultaneously exists at every point in space. You are using a pop-sci explanation of Schrodinger's cat that indeed falls apart when you dig deeper.$^*$ The key point is that a system cannot be in multiple states at once.
Schrodinger's cat (or if you hate this example, think "quantum system") is always in a single state. Typically the example says that there is an equal probability of us "measuring" the cat to be either alive or dead once we open the box. Therefore, the cat is in a state that is a superposition of our "life states" $|\text{alive}\rangle$ and $|\text{dead}\rangle$: $$|\text{cat}\rangle=\frac{1}{\sqrt{2}}\left(|\text{alive}\rangle+|\text{dead}\rangle\right)$$
This state tells us that there is a probability of $0.5$ of observing the cat as alive and a probability of $0.5$ of observing the cat as dead. This is because $$|\langle\text{alive}|\text{cat}\rangle|^2=0.5$$ $$|\langle\text{dead}|\text{cat}\rangle|^2=0.5$$
Once we open the box (perform a "life state" measurement of the system), the state of the cat collapses to one of the life states (eigenstates of the "life measurement operator"). So we observe the cat as either alive or dead.
It is important to understand that before we open the box the cat is not both alive and dead. The system cannot be in multiple states at once. It is in a single state, and this state is described as a superposition of life states. Once we open the box the cat is in a new single state which is one of the two life states. We cannot determine which state the cat ends up in though, only the probabilities it will end up in a certain state.
Of course Schrodinger's cat is crazy to think about because we are trying to apply QM formalism to the macroscopic world, but this is precisely how quantum systems work. We can express the state $|\psi\rangle$ of a quantum system as a superposition of eigenstates $|a_i\rangle$ of a Hermitian operator $A$: $$|\psi\rangle=\sum_ic_i|a_i\rangle$$ We do not say that the system is in every state $|a_i\rangle$ at once. It is in a single state (the superposition) that tells us the probability $|c_i|^2$ of the system being in one of the states $|a_i\rangle$ after making a measurement of the physical quantity associated with operator $A$.
$^*$I will use the Copenhagen interpretation of QM for my answer, since it is the most widely used interpretation to teach introductory QM. This is just one way to view this thought experiment, and it certainly is not a complete explanation. There are other interpretations that get to deeper meanings, more practical understand of measurements, etc. For that I'll refer you to the other answers, but I am not claiming this is the only way to view this scenario or QM in general. This question is not asking for a full explanation of the Schrodinger's cat experiment with a look into the deeper meaning of QM, so I am not going to get into all of that. The main point of this answer does not depend on the QM interpretation anyway.
Basically the answer is yes, the cat is both dead and alive. People used to discuss this sort of thing in terms of the Copenhagen interpretation (CI) and the Many-Worlds interpretation (MWI), but those discussions tend not to be satisfying, because both CI and MWI are designed so that in almost all real-world measurements, they give the same predictions. A better way to talk about this is in terms of decoherence.
Quantum mechanics says that the cat is in a superposition of states, alive and dead. Quantum mechanics doesn't impose any maximum size on objects that can be in a superposition of states. Double-slit interference has been observed with large molecules https://arxiv.org/abs/1310.8343 , and there are serious proposals to do it with a virus: http://arxiv.org/abs/0909.1469
However, due to interaction with its environment (e.g., vibrations from the walls of the box, and infrared radiation), the definite phase relationship between the live and dead parts of the cat's wavefunction would be lost very rapidly -- the time scale for a cat in a box would be many orders of magnitude too short to allow us to do anything during that time. Once the phase information is effectively lost, it becomes impossible to observe wave interference effects between the live and dead cat.
We humans aren't THAT important. Things happen whether we see them or not.
Right, this was always one of the unsatisfactory things about CI. Decoherence actually happens regardless of whether we observe the object at all. Our interaction with the system would cause decoherence, but so do other interactions, and they do it on much shorter time scales.
I can only consider it a fundamental breakdown of seemingly intelligent minds.
Lots of things in physics are counterintuitive.
I feel like all the answers here are missing the point.
The cat is not both alive and dead at the same time. That would be, as you put it, ludicrous. The truth is that the cat is in a superposition state of the states "alive" and "dead".
The problem is that there is no way to make sense of this statement without studying the underlying mathematics. Humans have no intuition for the concept "superposition", but some very smart people have found out that this concept describes our reality.
When scientists are asked to describe the experiment in layman's terms, they cannot say "you have to study the underlying mathematics". So they make their best effort to appeal to the layman's intuition by saying that the cat is both alive and dead at the same time. This is of course wrong, but there is simply no better way to phrase it in layman's terms.