The Elizabethan period spans her coronation in 1558 through her death in 1602.
Elizabeth was born in 1533, child of King Henry VIII and his second wife Anne Boleyn. King Henry had previously married Catherine of Aragon. While Catherine gave Henry a daughter, Mary, she did not give birth to a surviving male heir. In Catholic England, divorce was not permissible but because Catherine had briefly been married to Henry's elder brother Arthur, a Papal dispensation had been granted to permit the marriage in the first place. Henry wanted this dispensation to be ruled invalid thus invalidating the marriage and permitting him to marry Anne. The Pope stalled on the King's Great Matter until Henry broke England away from Rome: creating Protestant England, absorbing Papal property, and granting himself a divorce in the process. This made Elizabeth, the issue from this second marriage, a controversial claimant to the throne.
In the event, Henry sired a son of his third wife Jane Seymore. Edward VI would take the throne upon Henry's death. A sickly child, Edward died at 15, producing a small scuffle in the 9 day reign of Lady Jane Grey. Mary was the subsequent Queen and upon her death Elizabeth took the throne.
The biggest direct threat to the throne was Mary (Stuart), Queen of Scots. Mary was the great-granddaughter of Henry VII and niece of Henry VIII. Margaret Tudor, Henry VIII's older sister, married James IV of Scotland. The Catholic interpretation of events held that (protestant) Elizabeth was illigitimate and as her (catholic) half-sister Mary I was dead, the next in line was Mary Stuart. Mary was briefly the Queen of France, but Francis II died very young and Mary returned to Scotland. She was forced to abdicate as Queen of Scotland in favor of her son James following her second husband's murder. She fled to England, where she was again imprisoned, this time by Elizabeth.
In the 20 years of Mary's imprisonment, she was the convenient focal point of many plots. One was to marry her to Thomas Howard, the Duke of Norfolk -- this led to Norfolk's execution. Elizabeth is believed to have considered marrying her off to Robert Dudley (bringing the Scots under English rule). Mary was executed for treason after being set up (Babbington Plot). After Elizabeth's death, Mary's son, James VI of Scotland became James I of England.
Another threat was the balance of power with Spain and France. France supported Mary. Ireland was Catholic.
Queen Elizabeth ruled England, maintaining her power through careful manipulation of the Church and the feuding powers of Robert Dudley, the Earl of Leicester and Lord Burleigh. Elizabeth's favor (and the power and prestige thereof) were sought by the players at Court.
Elizabeth's tightrope took her through reformation of the Henrician church to a more protestant form. Henry VIII had altered the church to be his own branch of Catholicism. By breaking from Rome, he annexed the Churches lands in England. His son Edward (the Court in Edward's name) reformed the Church to be more protestant, and Elizabeth continued the trend. Catholics were upset at this but the protestants found her too moderate. Catholics were taxed in order to continue their faith and this became a symbol of prosperity among some of the aristocracy. This, coupled with Elizabeths power plays, antagonized the extreme protestants who bore the more familiar name of Puritans. The distress of the Puritans at the uncertain position of the Church lead many to emmigrate to New England. The true Catholics had grudges against Elizabeth primarily due to her Protestant heritage, making her illigitimate; these Catholics saw Mary Queen of Scots as the true successor to the throne.
Following Leicester's death, his feud with Burleigh became that of Essex vs Robert Cecil, Burleigh's son.
Leicester, Robert Dudley, whose father and grandfather were both executed, married Amy Robsart on the 4th of June 1550. They were both 17 years old , and the marriage is considered to have been a love match (at least in the beginning). Amy was the daughter of "an obscure country knight", and Robert therefore seems to have married "beneath" himself. Many years later, Amy was I believe found dead and it seems like Robert was suspected to have murdered her, so that he could marry Elizabeth instead. As you and I know, he never did. He choose an "easier" match instead, marrying the wealthy widow Lettice Knollys. Robert Dudley died in 1588.