I'm pretty sure that you mean the Battle of Bosworth, 1485. I'd actually say that this is one of the more famous battles in English history. In fact in terms of battles on English soil, I'd argue that only Hastings is definitely more famous. Bosworth was traditionally used to mark the end of medieval English history, and the beginning of early modern English history. It sees the death of the last Plantagenet King of England, Richard III, and the ascension of the first Tudor monarch, Henry VII.

I'll do my best to keep this fairly simple and brief as you can easily go into endless detail when explaining the background to this battle. Mid- to late fifteenth-century England saw a series of brief conflicts now known as the Wars of the Roses. The conflict was dynastic, essentially being fought over which family should rule England (plus its Welsh and Irish dominions). Henry VI was of the House of Lancaster, because he was descended from the Duke of Lancaster, Edward III's third-born son. He was deposed by members of the House of York, who had paternal descent from Edward III's fourth-born son (the Duke of York), but also had maternal descent from Edward III's second-born son. But this wasn't simply a disagreement over whether maternal descent counted. It was essentially a situation where two men had a claim to the throne, and if you didn't like one of them, you backed the other. If you supported the winner, they might reward you with land, money, and titles. If you supported the loser, the winner might confiscate your lands, or even have you executed.

At Bosworth, the Yorkist claimant was the ruling king, Richard III. His opponent was Henry Tudor. Richard undoubtedly had a better claim to the throne, since Henry was descended from the Duke of Lancaster (John of Gaunt) through the female line, and a bastard line at that. However, Richard had alienated a lot of people who had previously supported the House of York. He had usurped the throne from his young nephew, and was widely assumed to have had the boy murdered (see Dominic Mancini's first-hand account). He had replaced powerful men in the South with his supporters from the North. There had already been an unsuccessful rebellion against him, and the survivors had fled to join Henry Tudor in France. On 7th August 1485, Henry landed in South Wales with his English exiles and a force of French soldiers. Richard responded by ordering troops to be raised throughout the country, most of them by powerful lords. Relatively few, however, answered the call. Not only was there a risk of dying in battle, but lords also knew that they would be punished if they joined the losing side.

One powerful lord who did turn up was Lord Stanley who, aside from the King, was the wealthiest landowner in England. Incidentally, he was also King of the Isle of Man, although he didn't use the title much. He had broadly supported Richard III throughout his reign, but he also happened to be Henry Tudor's stepfather. Instead of joining his thousands of soldiers with Richard's main army, he deployed, as you say, 'on the sidelines'. In theory, he was there at Richard's summons, but according to Polydore Vergil he had promised Henry he would fight for him. Lord Stanley had a history of carefully waiting to back the winner, but in this case he had another reason to be cautious: Richard was holding Stanley's son hostage.

Richard's vanguard was held at bay by Henry's troops, led by the Earl of Oxford. In response, Richard ordered the Earl of Northumberland to bring his troops against those of Oxford. Northumberland failed to do so, either because the terrain made it impractical, or because he had decided to sit the battle out, waiting to see who won. Richard was in trouble, but he spotted an opportunity. Henry Tudor was, at this point, located away from his main army and protected by only a small force. Richard quickly led a small mounted charge against Henry, knowing that if he killed the pretender, the battle would be won. This cavalry charge, however, left Richard just as exposed as Henry. At this point, Lord Stanley's younger brother, Sir William Stanley, brought most of the Stanley troops into the battle. They surrounded Richard and killed him. With the King dead, Richard's army fled the field. It would probably be wrong to say that the Stanleys simply joined the winning side at Bosworth. Were it not for their decisive intervention, Richard would probably have killed Henry and thus won the battle. Rather than joining the winner, they chose the winner.

Our primary sources for the battle are relatively limited. The best is probably Polydore Vergil's account: http://www.philological.bham.ac.uk/polverg/25eng.html (the part on the Battle of Bosworth is at the bottom of the page). It is worth bearing in mind, however, that Polydore Vergil's employer was Henry VII. In recent decades, archaeology has added to our understanding of the battle.