Following a dispute in the 13th century over hunting rights on the land which was owned by the church, agreement was reached that the church owned the soil, but Madrilenos owned everything above the ground - namely game. Hence the symbol of Madrid was born - a Bear (the church's emblem) sniffing a tree. This bear can be seen across the city today, emblazened on taxis, buses, pavements, bins and almost everything belonging to "the city".
http://www.pbase.com/colinb3469/madrid_y…
El Oso y el Madroño (The Bear and the Strawberry tree)
Bears were plentiful around Madrid - herein probably lies the origin of Madrid 's shield El Oso y el Madroño (The Bear and the Strawberry tree - a favourite morsel of bears in Spain ). The area around Madrid was known as Ursaria for some time. The Libro de Montería (Book of the Hunt by King Alfono XI states: Madrid , un buen lugar de puerco y oso ( Madrid a good place for swine and bear).
http://www.iberianature.com/material/spa…
What is certain is that, from that time, the coat of arms of the local clergy continued the same as the old one of the city: a bear rampant with four paws grazing, on a field of silver, which we now know was the emblematic standard of the men of Madrid on the fields of Las Navas de Tolosa.
The City Council, however, modified its insignia or emblem, perhaps to make a statement on their new and now legitimate properties, for it depicts a bear on foot, on hind legs, eating fruit from the branches of a tree. Such it is as we know it today.
http://www.nova.es/~jlb/mad_in94.htm
There are several theories regarding the origin of the name, "Madrid". Madrid was founded by Ocno Bianor (son of King Tirenio of Tuscany and Mantua) and was named "Metragirta" or "Mantua Carpetana". Others contend that the original name of the city was "Ursa" ("bear" in Latin), due to the high number of these animals that were found in the adjacent forests, which, together with the Madrone tree ("madroño" in Spanish), have been the emblem of the city from the Middle Ages
http://en.wikipedia.org/wiki/Madrid
Directly opposite this building is the most important of the 3 statues to be seen in the square, "El Oso y El Madroño" (the Bear and the Strawberry Tree). Official symbol of the city (and as such, chosen as the background to ¡go Madrid!'s web pages), this bronze statue is the work of the sculptor Antonio Navarro Santa Fe. A reproduction of the Mariblanca statue (the original is in the Municipal Museum) and a statue of Carlos III (placed here by popular demand) can also be found in the square.
Lots of information from different view points
Anna