The first law of thermodynamics is basically that you can't get more energy of a system than you put in. Another way to express it is to say that the change in the energy of a system is equal to the work going in minus the heat coming out. There are actually several ways to describe the first law, so don't be surprised if your physics or science book list something completely different.
The second law of thermodynamics is basically that the entropy, or measure of disorder, in a system tends to increase. This doesn't mean that the entropy of a system can't decrease, but it does mean that if the entropy of a system decreases, the entropy of a larger system much increase by an equal or larger amount. This means that no matter what you do or what happens, the entropy of the universe is always increasing.
The third law of thermodynamics is usually stated as the inability for the temperature of a system to drop below absolute zero, 0 Kelvin. Temperature is typically described as the average kinetic energy of the particles that make up an object, often visualized as the particles vibrating; as the temperature increases, the vibrations increase. This means that absolute zero would be considered the temperature at which that vibration ceases. But due to quantum fluctuations, and the consquences of the Heisenberg uncertainty principle, particles can never stop vibrating completely, so it is considered impossible to reach, much less exceed, 0 Kelvin, hence the joke.
By the way, not long after this comic went up, a paper was published in which the authors claimed to have dropped the temperature of a system below absolute zero, but in my opinion, the claim was a bit facetious, since it was really more of a statistical trick than anything else.