A standard measure for pilots and air traffic control to refer to an aircraft's vertical position.
An aircraft's vertical position is measured with an altimeter, which compares a reading of the air pressure outside the aircraft to the known air pressure at a specific reference point. As a general rule the higher you get the lower the air pressure gets, so it can be used to work out vertical position.
Unfortunately air pressures change all the time and often vary over wide areas (see any weather report for evidence of this), compounded by the fact that a flying aircraft is not known to stay in one place. This means that as an aircraft flies through areas of different air pressure, its altimeter reading may fluctuate even if its altitude is constant (or more likely, the aircraft may climb or descend as a result of the pilot trying to maintain a constant altitude).
To maintain altimeter accuracy, an aircraft pilot can contact air traffic control to find out the regional air pressure at ground level or sea level (depending on the circumstances) and reset their altimeter accordingly.
This is fair enough for short hops in light aircraft but for jetliners, flying perhaps thousands of miles every day, it is clearly impractical; tens of altimeter resets would be needed every flight to maintain altimeter accuracy. Also the fact that the altimeters of many aircraft would never quite match up means a potential for aircraft to get too close to one another. In the UK, aircraft flying less than 3 or 5 miles apart must be at least 1000ft apart vertically, though all of these limits vary depending on the airspace.
The solution to this was the adoption of an international standard altimeter pressure setting of 1013.2 millibars, or 29.92 inches of mercury (for US altimeters). Incidentally this is also the International Standard Atmosphere (ISA) value for the air pressure at sea level, or zero feet. All aircraft above a specified altitude - called the 'transition altitude' (3,000 or 6,000ft in the UK, 18,000ft in the US) - set their altimeters to this standard pressure setting and after doing so, refer to their vertical position as their 'Flight Level'.
An aircraft's Flight Level is described as its level in thousands of feet measured against the ISA pressure for sea level, with the last two digits removed. For example, 'flight level two four five', or FL245, is approximately 24,500ft. 'Flight level nine zero', or FL90, is approximately 9,000ft. All flight levels are spoken this way except in the UK, in which flight levels ending with two zeros (i.e. FL100, FL200, etc) are spoken as 'flight level <number> hundred' to avoid confusion with the next level up. "Flight Level one hundred," for example, is more easily distinguished from "Flight Level one one zero" (the next level up) than "Flight Level one zero zero." The three-digit format allows flight levels to include hundreds of 'feet', though they are always given in either whole thousands or whole thousands plus 500ft. FL245 is a valid flight level, FL247 is not.
Of course flight level is not a true measure of altitude: unless the aircraft is in an area where the local air pressure at sea level is 1013.2 millibars, the altimeter will give a reading of more or less than the true altitude. However since all aircraft above the transition altitude are using the same altimeter setting, it doesn't matter whether their altitude readings are totally accurate or not, since they will all have the same inaccuracy. For separation purposes the vertical distances between aircraft is more important than their distances above ground, unless they're low enough that you have to start worrying about terrain clearance.
There will obviously be a point at which altitudes are referenced and the point at which flight levels are referenced meet. Since altitudes and flight levels are virtually always being measured against different datums (unless the local air pressure at sea level happens to be 1013.2 millibars), there is a gap between the transition altitude and the lowest available flight level called the 'transition layer'. Since flight levels are separated by 500ft, this layer will never be greater than 500ft 'deep' otherwise the lowest available flight level would roll over to the next lowest one.
Below the transition altitude, vertical distance is spoken as 'altitude' (distance above sea level) or 'height' (distance above ground level) depending on the situation, and is spoken as thousands of feet.
Occasionally pilots from countries with different transition altitudes confuse the terms - for example if a pilot in UK airspace was at FL140 but reported their altitude as 'one four thousand' instead of 'flight level one four zero', they would have to be corrected by air traffic control to ensure they were using the right altimeter setting. The highest transition altitude in the UK is 6,000ft; all aircraft flying above that should be using an altimeter setting of 1013.2 millibars and using the term 'flight level', not 'altitude' or 'height'. There have been instances where such confusion of terms has caused a loss of minimum vertical separation for aircraft.
Source: Duke, Graham; "Air Traffic Control, Eighth Edition" chp 3. "Radio Telephony"; ISBN 0-7110-2799-4