Short answers, pulled from the story.
The Palace of Westminster site has been in use since at least around 1045, when Edward the Confessor built a palace there. The current building dates from 1840, when construction began after the 1834 fire destroyed most of the original palace; it took thirty years to complete.
The Palace of Westminster is commonly called the Houses of Parliament because it houses the House of Commons and the House of Lords, the two legislative chambers of the UK Parliament. The term "Westminster" has also become a broader metonym for the UK Parliament and British Government.
The fire broke out on the 16th of October 1834 when an overheated stove used to destroy the Exchequer's stockpile of tally sticks set fire to the House of Lords Chamber. The blaze destroyed both Houses of Parliament and most of the palace complex, though Westminster Hall was saved.
The current Palace of Westminster was designed by architect Charles Barry, who chose a Perpendicular Gothic Revival style. Barry was a classical architect who relied heavily on Augustus Pugin to handle the Gothic decorative details; Pugin later expressed dissatisfaction with the symmetrical layout Barry imposed.
Big Ben is the popular name for the Great Bell housed in the Elizabeth Tower at the northern end of the Palace of Westminster. The tower was renamed from the Clock Tower in 2012 to mark the Diamond Jubilee of Elizabeth II. The bell weighs 13.8 tonnes and is the third-heaviest bell in Britain.
The palace was struck by bombs on fourteen separate occasions during the Second World War. The worst raid occurred on the night of the 10th-the 11th of May 1941, when at least twelve hits killed three people and destroyed the Commons Chamber. Westminster Hall was saved when firefighters prioritised it over the chamber. The rebuilt Commons Chamber was opened by King George VI on the 26th of October 1950.