An ideal gas is a theoretical matter state used by physicists in probability theory analysis. Ideal gas is made up of molecules that bounce off of each other without otherwise interacting at all. There are no forces of attraction or repulsion between the molecules, and no energy is lost during the collisions. Ideal gases can be completely described by their volume, density and temperature.
The state equation for an ideal gas, commonly known as the ideal gas law, is PV = NkT. In the equation, N is the number of molecules and k is Boltzmann's constant, which is equal to about 1.4 x 10^{-23} joules per kelvin. What is usually more important is that pressure and volume are inversely proportional, and each is proportional to temperature. This means, for example, that if pressure is doubled while temperature is held constant, then the volume of the gas must halve; if the volume of the gas is doubled while the pressure is held constant, the temperature must also double. In most examples, the number of molecules in the gas is considered to be constant.
Of course, this is just an approximation. Collisions between gas molecules are not perfectly elastic, some energy is lost, and electrostatic forces between gas molecules do exist. But in most everyday situations, the ideal gas law closely approximates the actual behavior of gases. Even if it is not used for performing calculations, keeping the relationships between pressure, volume and temperature in mind can help a scientist to understand a gas' behavior intuitively.
The ideal gas law is often the first equation that people learn when studying gases in an introductory physics or chemistry class. Van der Waal's equation, which includes a few minor corrections to the ideal gas law's basic assumptions, is also taught in many introductory courses. In practice, however, the correction is so small that if the ideal gas law isn't accurate enough for a given application, the Van der Waal's equation won't be good enough either.
As in most of thermodynamics, ideal gas is also assumed to be in a state of equilibrium. This assumption is clearly false if the pressure, volume or temperature is changing; if these variables are changing slowly, a state called quasi static equilibrium, the error may be acceptably small, however. Giving up the assumption of quasi static equilibrium means leaving thermodynamics behind for the the more complicated world of statistical physics.