Usually garlic is harvested when the bulb has formed into cloves, and is cured so that it will keep in storage. Green garlic (also known as spring garlic, baby garlic or new garlic) is picked when it's immature, the cloves haven't yet formed and the stalks are green and slender.  It's either pulled when crops being thinned or, increasingly, grown as a crop in its own right.
Looking rather like spring onions, with deep green stalks about 30 cm long, and pale, unsegmented bulbs, it differs from them in the purplish tint that sometimes appears on the stalk, and the shape of the leaves, which are flat, not tube-shaped like those of spring onions.
If allowed to grow past the 'spring onion' stage, but not to full maturity, green garlic produces a garlic 'round', a bulb like a boiling onion, still unseparated into cloves.
Nutrition & health  
Though the taste of green garlic is still recognisably garlicky, it's sweeter and more delicate than mature garlic. As with leeks, the entire plant is usable, and can be eaten raw or cooked in a wide variety of dishes.  Unlike mature garlic it can't be cured for storage should be kept in a refrigerator and used as soon as possible.
Similar claims have been made for its curative and preventative effects  as have for onions, especially in protecting against some cancers and lowering cholesterol. The juice has strong antiseptic properties and was used in this way by the French army in the First World War.
Possibly originating in the Kirghiz desert of Central Asia, garlic has been cultivated in Mediterranean countries since earliest recorded times. Workers building the Great Pyramid of Cheops were given a garlic ration to build their strength and ward off epidemics; it was popular with both ancient Greeks and Romans and in 1st-century India believed to help with heart and circulatory conditions. It's been known in Britain since at least the early 16th century.  
In recent decades the use of clove garlic has increased hugely in the UK, probably as a result of the boom in continental holidays; but the use of green stems is more traditional in Eastern cookery, so perhaps the current popularity of this ingredient here can be traced to the spread of oriental cuisines.