Lady Jane Grey
When Edward VI died, there was an obvious heir, his elder sister Mary. The problem was that she was Roman Catholic and Henry VIII had made England Protestant with the monarch as Head of the Church (and been excommunicated for his trouble). Mary, it was (accurately) thought, would reverse that and might well decide to persecute the Protestants in the process of turning to country back to Rome.
An alternative monarch was needed. So, four days after Edward's death, Jane Grey was proclaimed queen, with her husband, Guildford Dudley as her consort. Nine days later, and backed with troops, Mary arrived, imprisoned Jane and Guildford, and took the throne.
There was, however, a rising in favour of Jane, which cost the Nine-Day's-Queen her head.
The whys and wherefores
The basis of "Jane's" claim to the throne (there is no evidence that she ever wanted to be Queen) was that her mother, Frances Brandon, was daughter of Mary, who had been sister to Henry VIII, and that Henry had mentioned her in his will as a possible heir, should his son, and both of his daughters, Mary and Elizabeth, die without issue.
Frances Brandon forewent her superior claim to the throne, in favour of her daughter, at the urging of John Dudley (Guildford's father, Duke of Northumberland, minister to Edward, and prominent Protestant, whose head was likely to be high on Mary's list). Dudley had already tried, for a brief period, to put Jane forward as a bride for Edward VI, but the plan came to nothing. Dudley married Jane to his son, her sister Katherine to Lord Herbert, and his daughter Catherine to Lord Hastings (all at the one ceremony, with wedding garments borrowed from the Royal Wardrobe). He also tried to have Edward will the throne to her, but was blocked by the fact that Henry VIII had set down the succession in an Act of Parliament which placed Mary and Elizabeth ahead of her.
Jane is believed to have had the death of Edward kept from her, and to have renounced her claim in Mary's favour, but Dudley went ahead and had her declared queen. She was taken to the Tower of London to be crowned, and never left its walls again. She was pressured to accept the crown, and the first thing that happened after that was Guildford insisting that he be made king, something Jane refused to do.
Jane never received the support of the common people (she never had the chance to meet them) and Dudley and Guildford's high-handed attitude soured feelings towards them and, by association, to the helpless child behind whom they were machinating.
As for Mary I, she had Dudley executed, but contented herself with simply keeping Jane and Guildford in the Tower. Until, with news of her impending marriage to Philip of Spain, heir to the Hapsburg Empire (and a Catholic), dissent rose in the country, revolts were threatened, and Mary's Council (all of them Catholic) felt Jane's death essential. Even then the Queen offered Jane an escape, by sending a priest to her to suggest conversion to Catholicism. After hours of argument, she remained Protestant, and went to her death with dignity, even watching her husband's execution. She was calm on the scaffold, admitting publically that she ought never to have accepted the crown. She was buried at St.Peter's-ad-Vincula, alongside Anne Boleyn and Catherine Howard.