The Joust was a part of the Medieval
Tourney, or Tournament
. The origins of the tournament as entertainment are somewhat unclear, but it is generally accepted that it originated in the 11th century with the Normans
as a way of training knights and keeping them fit and occupied in times of peace.
A Joust consists of two men in full armour riding war horses and holding long wooden lances. They ride at each other from opposite sides of the field (or, later, a dedicated pitch called the lists) with the aim of unseating their opponent by hitting his shield with their lance. The popular image of a knight unseating his opponent and then jumping to the ground to finish him off is probably apocryphal, - among other things, it's not easy to jump in full plate armour - but on-foot confrontations did occur, usually as a way of determining the outcome of a very evenly matched fight.
Towards the later Middle Ages and into the Rennaissance the scoring system of the joust had evolved and become quite complicated. Though in the beginning the winner of any confrontation was probably declared by the lord hosting the tourney, in later times there were precise scores for every eventuality of the run. The highest score was given to a knight who managed to unseat his opponent, next came the shattered lance, then a lance that was broken at the tip. Points were deducted for hitting the horse or for pointing the lance at the head of the other rider. Thus it is easy to see how, riding three or more times against each other, two knights might reach a stalemate in points which would require on-foot resolution.
Originally the joust was a rather minor section of the tournament - it was preformed in the mornings, with the main entertainment, the melee or free for all (a simulation of group infantry and cavalry battle), as the main even in the afternoon. However, the relative safety of the joust and the opportunity it provided for individual nobles to distinguish themselves gradually helped it become the focal point of a tournament, with kings and princes taking part alongside the rank and file of chivalry (there is a record of Henry VIIIbeing hurt during a joust, bless him).
Contrary to the image presented by the otherwise excellent, but by no means historically accurate movie A Knight's Tale, there were usually no prizes for the winners in tournaments. They fought for glory and the admiration of the ladies, as well as to gain favour and honour in the eyes of prospective noble patrons. It was, however, possible to make money on the tournament round. It was at one time a seasonal circuit where poor knights and second sons (as well as enthusiasts hungry for fame) traveled from place to place fighting in tourneys and gradually amassing a fortune. A knight who bested another was entitled to his adversary's horse and armour, which would usually be ransomed back for a large sum of money. War horses and full suits of armour were not cheap - they probably represented what would be five year's salary of an average person today!
During the late 12th, 13th and early 14th centuries the ideas of chivalry, courtly love and romantic ideas of honour contributed to the laws and traditions of the joust evolving and of the tournament becoming an important part of aristocratic social life. Ladies granted "favours" - usually ribbons or other tokens worn on the sleeve in battle - to chosen champions who were thus dedicating their performance to their lady and bringing honour to her name. I could not find any accounts of whether or not these formal but public relationships ever clashed with contemporary mores of female modesty. People being what they are, however, and knowing that the church strongly dissaproved of tourneying practically from its inception, I rather imagine that well bred young ladies made double sure that their noble papas or husbands did not find out which bits of clothing they bestowed on which dashing combatant - unless of course said papa had an interest in his daughter showing favour to a particular marriage prospect.
Outside of tournaments jousts were used, in accordance with the laws of the time, as a method of judicial resolution. A person charged with any manner of crime, from treason to murder, could issue a challenge to their accuser after which there would be a one-on-one battle either between the accuser and accused themselves, or through appointed champion - on either or both sides. This type of joust was not awarded points and was not limited to a certain number of runs. It was fought until one side pleaded surrender or was killed or seriously wounded. The thinking at the time was that God would support the arm of the righteous man, and therefore whichever side won the joust was shown by devine power to have the right of the case. This type of combat later evolved into the duel, which persisted as a legal means of resolving questions of honour between men until the 19th century (the law was repealed in England in 1818).
The joust and its romantic splendour are enjoying a mass revival at the moment. Scores of groups and societies in both the UK and the US engage in historically accurate jousting (as well as the more tourist driven phony stuff) at fayres and re-enactment events. A short search on Google will return dozens of results, one of which might be for a group near you which can either display their stuff or accept you as a member and help you train in the noble art of jousting.