Properties and uses
Wild carrots, related to parsnips, parsley and celery, are native to southern Europe and Asia; domestic carrots were bred from them to give a larger, tastier, and less woody taproot.
Carrots contain exceptionally high levels of beta-carotene, converted by the human body into vitamin A, a deficiency of which can cause sight defects and blindness. 
Raw carrots release only 3% of their beta-carotene during digestion, while pulping, cooking or adding oil can improve this to 39%. So drinking carrot juice is probably the way to get most nutrients from raw carrots, but it's debatable whether raw or cooked carrots are better for us. Boiled or steamed carrots are easier to digest, so though some nutrients are lost in cooking, others are more easily absorbed. Boiling and steaming certainly preserve antioxidants better than other cooking methods, especially if carrots are boiled whole, without pre-slicing.  A study at Newcastle University found that boiling carrots whole dramatically increased their nutritional content, including that of the allegedly cancer-preventing compound falcarinol. They're are also rich in vitamins B and C, and minerals including iron, calcium, sodium and potassium.
They have a long record as a folk remedy for everything from arthritis to anaemia, including boils, burns, constipation, diarrhoea, dropsy, gout, rickets and threadworm. Once thought to be aphrodisiac when stewed with sugar, they've also been used in poultices to soothe exzema, herpes and chilblains.
The many varieties of modern carrot are usually classified by root shape - long, stumpy, conical, cylindrical - but as there are also lighter and heavier cropping, early, late  and over-wintering varieties, a range of carrots is now available all year round.  Carrots last longer with their tops removed, and keep for about a week in the fridge.
Ancient carrots weren't orange - they were white or purple, and easily confused with parsnips; carrots were given the name Daucus (supposedly by the 2nd-century physician Galen) to differentiate.  Like their relatives fennel, dill and cumin, early carrots seem to have been used as much for their aromatic leaves and seeds as for their roots. Ancestors of the modern carrot were introduced (or possibly reintroduced) to Europe from Afghanistan by the Moors in the 8th - 10th centuries, via north Africa and Spain. 
By the 16th century carrots had reached England. They were a favourite of Queen Elizabeth, while at the court of her successor James 1, ladies decorated their hair, hats, and dresses with the feathery leaves. Improved strains of three varieties, yellow, gold and red, were deveolped in 17th-century Britain, where carrots were now grown as a farm crop and eaten as an accompaniment to boiled beef. Meanwhile in Holland long, orange cultivars, prototypes of modern varieties, began to be produced (the colour - according to Dutch folklore - making them popular emblems of the House of Orange).
In the 18th and 19th centuries the carrot's natural sweetness led to wider use as an ingredient in puddings and cakes.  The "Dig for Victory" campaign of World War ll gave it a crucial role, providing children with nutrients no longer obtainable from fruit, though the notion that RAF crews had superior night vision from eating carrots was a myth put about by the RAF to mask their use of radar.