Correct spelling, explanation: legend is the correct spelling because of the Latin origin of the word (legenda). The original word is spelled with g, not dg so ledgend is a misspelling and legend is the correct form. In its first meaning, the word referred only to the life of saints, but later it has become wider.

Definition of legend:
noun, a story, usually folk, believed to be made up but with a pint of truth
According to the legend, everybody who looked the dragon in the eyes turned into a rock.
The weekend at the castle was wonderful, we learned an amazing legend about the place.

Phrases with legend:
urban legend – a story or anecdote based on hearsay that is told as a true one but probably is made up and repeated in many different versions and contexts. Usually scary or very controversial
The story about this murderer is just an urban legend – I’ve heard it about many different places!
a living legend – somebody famous who is considered a genius in their field
I think the Rolling Stones are living legends.

Incorrect spelling, explanation: this spelling is incorrect, as there is no trace of dg spelling back in the history of the word. Legend is the correct form and it comes from Latin legenda, later French legende – all forms written as later English legend. Therefore ledgend is an incorrect form.