Like most aspects of medieval society, it depended quite heavily on where and when and you lived. There's a whole debate about how knighthood originated, which essentially amounts to a question of whether the nobility of the Carolingian Empire (the Empire of Charlemagne and his successors which lasted from 751 to 888) was already defined by their warrior status - Janet Nelson in her essay "9th century knighthood: the evidence of Nithard" in her book "The Frankish world, 750 - 900" is a proponent of this view. Other historians such as Georges Duby in "the chivalrous society" and Jean Pierre Poly in "the Feudal transformation, 900 - 1200" see knighthood and the militarisation of the nobility as only really developing from the 10th century on. Certainly, not all knights were noble to begin with. For example, Germany under the Ottonian (919 - 1024), Salian (1024 - 1125) and Hohenstaufen (1155 - 1245) king-emperors you had ministeriales - serfs raised to knighthood who were entrusted with the governance of royal castles and various other military and administrative duties for the emperor (the fact that they serfs and so were still legally unfree meant that they were easier to control than freeborn nobles). Eventually, when Germany descended into a period of prolonged interregnum and civil war from 1245 to 1273, the ministeriales broke free, took control of the castles they'd been entrusted with and formed the German lower nobility (adel). On the other hand, the nobility seems to have embraced knighthood no later than the mid-11th century i.e. William of Poitiers in his "The Deeds of William" describes how the young William the Conqueror was knighted by King Henry I of France, and Orderic Vitalis describes how William the Conqueror had his youngest son Henry dubbed a knight. How closely associated with nobility and exclusive knighthood was continued to depend on where you were throughout the Middle Ages. In France, you had to show proof of nobility, ideally by having three noble grandparents, while in England any one who served the crown well-enough in battle could get knighted i.e. one of the greatest English knights and mercenary captains of the 14th century, Sir John Hawkwood, enrolled as a longbowman on the Poitiers campaign of 1355 - 1356 and was knighted for distinguishing himself there sometime after it, and Robert Salle, a knight killed in the peasants' revolt, was said to have himself been born a villein (unfree peasant). Indeed, in England, anyone possessing land worth more than £40 had to become a knight - if they didn't they'd have to pay a fine to the king. Italy was a different matter too, as the city-state republics, though they had their own ancient and hereditary warrior nobilities, would (especially during their early growth in the 12th and 13th centuries) bestow knighthood on many of the merchants, bankers and lawyers who irregularly came on military campaigns (service in the communal militia was a duty of all enfranchised male citizens) - the Florentine poet Franco Saccheti satirised this, claiming that if one was going to knight merchants, one might as well knight oxen and donkeys. Everywhere in Europe, however, the later you go into the Middle Ages the smaller the portion of knights gets (i.e. in 1356, 25% of English cavalrymen were knights but by 1415 only 8% were) and the more zealously knightly status is guarded.

As for how becoming a knight worked it really depended on who you were. As I said, some knights were raised from humble origins through service, but noblemen were trained for it from childhood. For example, Geoffrey Plantagenet (1113 - 1151), Count of Anjou, spent his childhood growing up with sons of lesser lords from his county, who had been sent to his castle to be fostered with him to strengthen loyalties between them, and together they trained as knights until they were all knighted together in a mass ceremony in 1128 at Rouen by King Henry I of England, Geoffrey's father-in-law, followed by a tournament in which Geoffrey won the best honours and a feast. The way a man would be dubbed a knight would work, as outlined in an old French poem the Ordene de chevalerie (c.1220), went like this:

1. the day before the dubbing, the knight would take a ritual bath so that he would both be physically clean for his ceremony and that the sins of his life up until then would be symbolically washed away.
2. He would then rest on a new bed dressed with clean linen, as if emerging from a fight against sin and Satan.
3. When he awoke he would dress for the ceremony in a red tunic (symbolising his willingness to shed blood in defence of the church and his lord), black hose (to symbolise his mortality), a white belt (to symbolise his continence and chastity) and a red cloak (to show humility).
4. After that, he would attend mass in the Church.
5. When the service finished, he would be given golden spurs and then the lord conferring knighthood on him would hand him a sword, kiss him and then tap him on the shoulder.

The best introductory work on this subject still remains "Chivalry" (1984) by Maurice Keen, so I'd highly recommend reading that to start with.