Warning: wikipedia contains spoilers
The play opens with the famous speech by Richard, beginning, Now is the winter of our discontent... The speech reveals Richard's jealousy and ambition, as his brother, Edward IV of England, rules the country successfully. With little attempt at chronological accuracy, Richard is shown ingratiating himself with "the Lady Anne" -- Anne Neville, widow of the Lancastrian Prince of Wales. Despite her prejudice against him, Anne is won over by his pleas and agrees to marry him. Richard, in collaboration with his friend Buckingham (Henry Stafford, Duke of Buckingham), plots to be the next king, and presents himself to the other lords as a modest, devout man with no pretensions to greatness. This causes them to select him as king after Edward IV's death, putting aside the claims of his innocent young nephews (the Princes in the Tower).
Richard's crimes go from bad to worse. He murders all who stand in his way, including the young princes, Lord Hastings, his former ally Buckingham, and even his wife. When he has lost all popular support, he faces the invading Earl of Richmond (Henry VII) at the Battle of Bosworth Field. Prior to the battle, Richard is visited by the ghosts of those whose deaths he has caused, all of whom tell him to Despair and die! Alone on the field at the climax of the battle, he utters the often-quoted line, A horse, a horse, my kingdom for a horse! He is defeated in hand-to-hand combat by Richmond, and dies dramatically.
Shakespeare's main source for his play was the chronicle of Raphael Holinshed. He may also have drawn on the work of Sir Thomas More, who was a protege of John Morton, one of Richard III's implacable enemies. More is supposed by some to have had a copy of Morton's manuscript of his version of the story, but this manuscript, if it ever existed, does not survive.
Richard III is the culmination of the cycle of Wars of the Roses plays. In Henry VI Part III, Shakespeare had already begun the process of building Richard's character into that of a villain, even though he could not possibly have been involved in some of the events depicted. From an overview of the cycle, it can be seen that Shakespeare's inaccuracy works both ways.
The most famous player of the part in recent times was Sir Laurence Olivier in the 1950s film version. His inimitable rendition has been satirised by many comedians including Peter Sellers (who had aspirations to do the role straight). Sellers' version of A Hard Day's Night was delivered in the style of Olivier as Richard III.
More recently, Shakespeare's Richard III has been brought to the screen by Sir Ian McKellen (1995) in an abbreviated version set in a 1930s fascist England, and by Al Pacino in the 1997 documentary, Looking for Richard.
(links are to articles on the historical personages, who may not precisely correspond to Shakespeare's portrayal of them)
King Edward IV
Edward, Prince of Wales, afterwards Edward V, son to the king
Richard, Duke of York, son to the king
George, Duke of Clarence, brother to the king
Richard, Duke of Gloucester, afterwards King Richard III, brother to the king
Edward, Earl of Warwick, young son of Clarence
Henry, Earl of Richmond, afterwards King Henry VII
Thomas Cardinal Bourchier, Archbishop of Canterbury
Thomas Rotherham, Archbishop of York
John Morton, Bishop of Ely
Duke of Buckingham (Henry Stafford)
Duke of Norfolk (John Howard, 1st Duke of Norfolk)
Earl of Surrey, his son Thomas Howard)
Earl Rivers (Anthony Woodville, 2nd Earl Rivers), brother to Queen Elizabeth
Marquess of Dorset(Thomas Grey, 1st Marquess of Dorset), son to Queen Elizabeth
Lord Grey, son to Queen Elizabeth
Earl of Oxford
William, 1st Lord Hastings
Lord Stanley (Thomas Stanley), afterwards Earl of Derby
Lord Lovel (Francis Lovell, Viscount Lovell)
Sir Thomas Vaughan
Sir Richard Ratcliffe[?]
Sir William Catesby
Sir James Tyrrel
Sir James Blunt (James Blount)
Sir Walter Herbert[?]
Sir Robert Brackenbury[?], Lieutenant of the Tower
Christopher Urswick, a priest
Hastings, a pursivant
Tressel and Berkeley, gentlemen attending on the Lady Anne
Keeper in the Tower
Lord Mayor of London (Sir Edmund Shaa)
Sheriff of Wiltshire
Elizabeth Woodville, Queen to Edward IV
Margaret of Anjou, widow of Henry VI
Duchess of York (Cecily Neville), mother to King Edward IV, Clarence, and Gloucester
Lady Anne Neville, widow of Edward, Prince of Wales (son of Henry VI), afterwards married to Gloucester
Margaret Plantagenet, Countess of Salisbury, young daughter of Clarence
ghosts, lords, gentlemen, citizens, etc.