Official abbreviation: R.E.

Instituted circa 1457

Awarded to the Danish Royal Family, Heads of state and other distinguished individuals.
The order consists of just a single class. When awarded the Order of the Elephant the recipient becomes Knight of the Order of the Elephant and gets the honorary title of Excellency (His/Her Excellency)

Other information: The order is almost never awarded to commoners. The latest commoner recipient who wasn’t a head of state or a royal, was HE Mærsk Mc-Kinney Møller, Former CEO of A. P. Moller-Maersk Group, former Member of the board of IBM and the richest man in Denmark with a personal wealth of 7 Billion DKK (1,200,000,000.00 USD). Only 3 danish commoners have received the medal in the last 100 years.
For more information, take a look at the english wikipedia.

Design: The Order of the Elephant consists of a sash to be worn over the shoulder, a necklace and a breast star.
The insignia consists of an elephant adorned by a tower.
The necklace is made from gold and is only worn on New Years eve or on the birthday of the reigning monarch.
Note that when only the ribbon is worn, it is an extra wide ribbon
The sash is about 10 CM wide and is worn over the left shoulder.
The star is worn on the left side of the breast.
The order is worn in chain three times a year: 1. January, at the monarch birthday (16. April for Queen Margrethe II) and on Valdemar Sejrs birthday 28. June.

Notable (non-royal) recipients:
Astronomer Tycho Brahe
Mærsk Mc-Kinney Møller (2000)
Physicist Niels Bohr (1947)
East Asia Company founder H.N. Andersen (1919)
Sir Winston Churchill
Dwight D. Eisenhower

Sash and breaststar
Official drawing of the necklace. Circa 1893.
Newer style metal breast star. The older breast stars are cloth.