The Zeroth Law of Thermodynamics

If two bodies are separate and in a thermal equilibrium with another body, then the two bodies will be in thermal equilibrium once placed in thermal contact with one another.

Alright . . . what does that mean? Say you have an ice pack, a hot charcoal, and a bucket of water. The bucket of water is probably in thermal equilibrium, meaning that the temperature of the water is the same everywhere inside the bucket. It's not like the top part of the water is really hot, and the bottom part is freezing cold. So you stick both the ice pack and the hot charcoal inside the water. After some time . . . the two objects will have the same temperature! But you already knew that right?

 
The 1st Law of Thermodynamics

This is basically the Law of Conversation applied to heat. Energy can be transferred by work done on a system, or by heat transfer. In order to show this mathematically . . . we have to specify variables for work and heat transfer. Usually work is denoted by W. Now we have to think of an object. Which is part of a system, and is surrounded by its surroundings. When work is done on the system it is negative, and when it is done on its surroundings it is positive. Heat transfer is known as Q. When heat is lost from the system it is negative, (-Q), and when it is added it is positive (Q). So we get this equation:

U = Uf - Ui = Q - W . . . --> do the word thing.
 
The 2nd Law of Thermodynamics

This law deals with heat engines mostly. It says that no heat engine can successfully convert heat into 100% work. There is always something lost in the process. Like exhaust from a car. So we use what is known as thermal efficiency. This is simply a ratio between the input energy, and the output energy.

Carnot Engine
STEP 1: Isothermal expansion - The system takes heat from the high temperature source and expands isothermally. Since there is no temperature increase all the heat taken from the source is put into work done by the gas.
STEP 2: Adiabatic expansion - The system can expand without exchanging heat with its surroundings. The internal energy and temperature decrease because the system expands, and performs positive work.
STEP 3: Isothermal compression - The system discards heat to the low temperature reservoir, because word is done on the gas by the environment.
STEP 4: Adiabatic compression - Now the system returns to it's initial state (Step 1). Since the system is compressed, the internal energy and temperature increase, because no heat is discarded.