Airspeed is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are indicated airspeed ("IAS"), calibrated airspeed ("CAS"), equivalent airspeed ("EAS"), true airspeed ("TAS"), and density airspeed. Indicated airspeed is simply what is read off of an airspeed gauge connected to a pitot static system, calibrated airspeed is indicated airspeed adjusted for pitot system position and installation error, and equivalent airspeed is calibrated airspeed adjusted for compressibility effects. True airspeed is equivalent airspeed adjusted for air density, and is also the speed of the aircraft through the air in which it is flying.  Calibrated airspeed is typically within a few knots of indicated airspeed, while equivalent airspeed decreases slightly from CAS as aircraft altitude increases or at high speeds. With EAS constant, true airspeed increases as aircraft altitude increases. This is because air density decreases with higher altitude, but an aircraft's wing requires the same amount of air particles (i.e., mass of air) flowing around it to produce the same amount of lift for a given AOA; thus, a wing must move faster through thinner air than thicker air to obtain the same amount of lift.
The measurement and indication of airspeed is ordinarily accomplished on board an aircraft by an airspeed indicator ("ASI") connected to a pitot-static system. The pitot-static system comprises one or more pitot probes (or tubes) facing the on-coming air flow to measure pitot pressure (also called stagnation, total or ram pressure) and one or more static ports to measure the static pressure in the air flow. These two pressures are compared by the ASI to give an IAS reading.
Indicated airspeed (IAS) is the airspeed indicator reading (ASIR) uncorrected for instrument, position, and other errors. From current EASA definitions: Indicated airspeed means the speed of an aircraft as shown on its pitot static airspeed indicator calibrated to reflect standard atmosphere adiabatic compressible flow at sea level uncorrected for airspeed system errors.
Outside the former Soviet bloc, most airspeed indicators show the speed in knots (nautical miles per hour). Some light aircraft have airspeed indicators showing speed in statute miles per hour or kilometers per hour.
An airspeed indicator is a differential pressure gauge with the pressure reading expressed in units of speed, rather than pressure. The airspeed is derived from the difference between the ram air pressure from the pitot tube, or stagnation pressure, and the static pressure. The pitot tube is mounted facing forward; the static pressure is frequently detected at static ports on one or both sides of the aircraft. Sometimes both pressure sources are combined in a single probe, a pitot-static tube. The static pressure measurement is subject to error due to inability to place the static ports at positions where the pressure is true static pressure at all airspeeds and attitudes. The correction for this error is the position error correction (PEC) and varies for different aircraft and airspeeds. Further errors of 10% or more are common if the airplane is flown in “uncoordinated” flight.
Calibrated airspeed (CAS) is indicated airspeed corrected for instrument errors, position error (due to incorrect pressure at the static port) and installation errors.
Calibrated airspeed values less than the speed of sound at standard sea level (661.4788 knots) are calculated as follows:
minus position and installation error correction.
Units other than knots and inches of mercury can be used, if used consistently.
This expression is based on the form of Bernoulli's equation applicable to a perfect, incompressible gas. The values for and are consistent with the ISA i.e. the conditions under which airspeed indicators are calibrated.
Equivalent airspeed (EAS) is defined as the speed at sea level that would produce the same incompressible dynamic pressure as the true airspeed at the altitude at which the vehicle is flying. An aircraft in forward flight is subject to the effects of compressibility. Likewise, the calibrated airspeed is a function of the compressible impact pressure. EAS, on the other hand, is a measure of airspeed that is a function of incompressible dynamic pressure. Structural analysis is often in terms of incompressible dynamic pressure, so that equivalent airspeed is a useful speed for structural testing. At standard sea level pressure, calibrated airspeed and equivalent airspeed are equal. Up to about 200 knots CAS and 10,000 ft (3,000 m) the difference is negligible, but at higher speeds and altitudes CAS must be corrected for compressibility error to determine EAS. The significance of equivalent airspeed is that, at Mach numbers below the onset of wave drag, all of the aerodynamic forces and moments on an aircraft are proportional to the square of the equivalent airspeed. The equivalent airspeed is closely related to the indicated airspeed shown by the airspeed indicator. Thus, the handling and 'feel' of an aircraft, and the aerodynamic loads upon it, at a given equivalent airspeed, are very nearly constant and equal to those at standard sea level irrespective of the actual flight conditions.
True airspeed is the speed of the aircraft relative to the atmosphere. The true airspeed and heading of an aircraft constitute its velocity relative to the atmosphere. The vector relationship between the true airspeed and the speed with respect to the ground is:
Aircraft flight instruments, however, don't compute true airspeed as a function of groundspeed and windspeed. They use impact and static pressures as well as a temperature input. True airspeed is equivalent airspeed that is corrected for pressure altitude and temperature (which define density). The result is the true physical speed of the aircraft relative to the surrounding body of air. At standard sea level conditions, true airspeed, calibrated airspeed and equivalent airspeed are all equal.
The simplest way to compute true airspeed is using a function of Mach number:
Or if Mach number is not known:
The above equation is only for Mach numbers less than 1.0.
True airspeed differs from the equivalent airspeed because the airspeed indicator is calibrated at SL, ISA conditions, where the air density is 1.225 kg/m³, whereas the air density in flight normally differs from this value.
The air density may be calculated from:
Source: Aerodynamics of a Compressible Fluid.
Ground speed is the speed of the aircraft relative to the ground, which is the aircraft's true airspeed combined with the speed of the air mass (i.e., wind) in which it is flying. For instance, an aircraft traveling 150 knots TAS in an air mass moving against its direction of flight at 25 knots would experience a headwind of 25 knots and have a groundspeed of 125 knots.