Phosphorescence is a type of luminescence.
Luminescence occurs when a substance emits light for any reason other than those related to a rise in temperature. For example, if wood is set on fire, it emits light, but this is caused by the process of combustion rather than being caused by luminescence.
Different Types of Luminescence
Luminescence is itself a broad term covering many different types of light emission that occur without heat.
Light emitted by a living organism, such as some deep sea creatures.
This is light emitted from a chemical reaction. This type of luminescence is used in glowsticks. Another form is electrochemiluminescence, caused by an electrochemical reaction.
This happens when light is emitted when some substances crystallize.
Electroluminescence is caused by an electrical current being passed through a substance, such as in the backlit dashboard of a car. Also includes cathodoluminescence, which happens when a substance is hit by an electron.
This is caused by mechanical actions on solids. This also includes triboluminescence caused when material bonds are broken by scratching, crushing or rubbing, fractoluminescence caused by the fracturing of bonds in crystals and piezoluminescence caused by pressure on a solid.
This is caused by the absorption and then re-emission of photons, and is divided into fluorescence and phosphorescence.
Caused when a substance is bombarded by ionizing radiation. Tritium in items such as gun sights use this type.
Sonoluminescence occurs when sound implodes bubbles in a liquid.
This is when absorbed light is re-emitted upon heating.
Phosphorescence, or persistent luminescence, is used to describe situations where the substance continues to emit light for a significant time after whatever caused the emission is removed (the "exciter"). Fluorescence is where the process does not continue for any significant length of time afterwards. The definition is a bit arbitrary as light emission always continues after the exciter is removed, although this is rarely noticeable to the unaided eye. Some definitions of phosphorescence consider it to be where the persistence is greater than 10 nanoseconds. 
Phosphors are substances that exhibit the properties of luminescence. This includes both phosphorescent and fluorescent substances. They are commonly transition metal or rare earth compounds, some of which are naturally occurring.
Phosphorescent phosphors are used in items such as radar screens, safety signs, watch faces, CRT displays and glow in the dark toys to name just a few uses.
The ability to absorb light and re-emit it later without any power is very useful in items such as safety signs, as phosphorescent signs can show where a fire door is without the need for power; useful as during a fire, power may well have been interrupted. They are also used in lighting applications such as fluorescent lamps and in neon signs, in the latter of which they are used to produce different colours.
Watch dials used to use radium to glow in the dark. This was not long after radioactivity had been discovered, and misleading information led to many deaths and illnesses from radiation poisoning, such as the infamous Radium Girls at the U.S. Radium Corporation's factory in New Jersey. The radium was later replaced with less harmful substances.
Some Common Phosphors
Zinc Sulphide Commonly used in glow in the dark toys and can come with different activators. Using copper as an activator gives a long glow time and the greenish glow common in glow in the dark stickers. Using silver gives a bright blue glow. Manganese gives an orange-red glow. Zinc sulphide, doped with either copper or silver, is probably the best known phosphor.
Strontium Aluminate A newer material with a europium activator, it glows in hues of green and aqua; green glowing brighter, and aqua longer. Overall, it is brighter and glows for longer than older materials such as Zinc Sulphide; it is also a lot more expensive.
Zinc Oxide With a zinc activator, it is commonly used in vacuum fluorescent displays (VFD) in consumer electronics such as microwaves and car radios as an alternative to liquid crystal displays (LCD).
Calcium Sulfide Using a strontium sulfide with bismuth as an activator, this gives off a blue light for up to 12 hours.
Zinc sulphide and calcium sulphide can be combined. The colour emitted by them depends on the ratio of each compound; by adjusting the ratios, the emitted colour can be altered.