Short answers, pulled from the story.
Cordelia tells King Lear "Nothing, my lord" when he asks what she can say to earn a share of his kingdom. She explains that she cannot heave her heart into her mouth, and that she loves him "according to her bond; no more nor less" (1.1.90-2). Her refusal to flatter him results in her banishment and disinheritance.
Cordelia is hanged in prison after she and Lear are captured following her punitive expedition against her sisters Goneril and Regan. Edmund orders them both to prison, and Cordelia is killed there. Her death is one of the most debated endings in dramatic literature.
The oldest source in print is Geoffrey of Monmouth's The History of the Kings of Britain, written around 1136. In that text, Cordelia appears as Queen Cordelia, not as a character who dies in prison. It is the earliest written record of the character that Shakespeare had available when writing King Lear.
In his 1681 revision, The History of King Lear, Nahum Tate removed the King of France from the story and had Cordelia marry Edgar and become ruler of the kingdom, surviving the play. Tate's version replaced Shakespeare's original on stage for decades.
The King of France marries Cordelia because he is impressed by her honesty, which had just caused her disinheritance. The Duke of Burgundy withdrew his marriage suit when he learned she had been disinherited; the King of France chose her for the very quality that cost her her inheritance.
Screen portrayals of Cordelia span more than a century, from Lorraine Huling in the 1916 film directed by Ernest C. Warde to Florence Pugh in the 2018 Amazon adaptation directed by Richard Eyre. Other notable actresses in the role include Brenda Blethyn in a 1982 television production directed by Jonathan Miller and Romola Garai in a 2009 PBS production directed by Sir Trevor Nunn and Chris Hunt.